In Progress: the one here is the one called "L. Last-Ditch Defense" in the main file, prefaced by 66. Kf6 Qb6+ 67. Kg7 Qe6 68. Qf3/f6.
68. ... Qc8!?
It is a paradox of this ending that Black willingly hangs his pawn with check: 69. Qxd5+? is a computer-proven draw. Now White's King is contained for the moment: 69. Kh7 Qc2! seems so good as to be an easy draw for Black, 69. Kf7 simply meets ...Qc7+ if not ...Qd7+, and 69. Kh6 Qh8+ 70. Kg5 Qe5+ 71. Qf5 Qe7+ is not great progress for White. Thus it is time for White to re-position his Queen with checks. Three target squares seem particularly inviting: e2, b4, and b6; the last of these seems the one to fear.
It is worth inserting Black's defenses to the first two, since they also show rich themes: 69. Qe2+ Ka1/b1 70. Qd1+ Ka2/b2 71. Qd2+ Kb1 72. Qb4+ is met by 72...Ka2! and on 73. Kf6 Qc6+ 74. Kf7 Qd7+ 75. Qe7 Qf5+ we transpose into positions White could have had long ago, in which Black is able to block White's King from going to the Q-side. The other line leads to pretty play if Black moves our King to b1: 69. Qe2+ Kb1!? 70. Kf6(!)(?) Qf8+ 71. Kg5 Qd8+ 72. Kf5! Qf8+ 73. Kg4 Qc8+ 74. Kg3 Qc7+ 75. Kf3! Here 75. Kg2 allows ...d4! since Black has a forking check at c6, but now that check will be met by the cross-check Qe4+ trading Queens. Of all Black's tries, the best seems to be the immediate 75...Qg7!, when 76. Qd3+ Kb2 77. Qf5 d4(!) 78. Qf7 Qh6! 79. g7 Qh3+ is a pattern the World Team found to be good in other lines with White's Queen at f7---Black can deny White a useful interposition, and 80. Ke4 Qe3+ 81. Kf5 Qf3+ 82. Kg6 Qg3+ 83. Kh6 Qe3+! 84. Kh7 allows Black both the defense 84...Qe4+ 85. Qg6 Qh4+ 86. Qh6 Qe4+ 87. Kh8 Qe5 that the misguided 58...Qe4? originally sought to achieve, and the probably better 84...Qh3+! 85. Kg8 d3! and Black seems to have caught up in the pawn race. Nor does 79. Kg4? d3! seem to work as Black is queening with check. Moreover, 69. Qe2+ Ka1 looks possible on account of 70. Qe5+ d4!, and this is a case where Black draws even if the pawn falls on d4 with check, and Black also appears able to avoid the dreaded "Pin Battery" idea (White running with his King so as to play Kxd4 and a discovered check) because the Queen on e5 is not guarding g6. But enough of second-best moves for White...
69. Qf2+! Ka1 70. Qa7+! Kb1 71. Qb6+! Ka2
Now the ONLY difference from White's option of playing 59. Qb6+ right back at the beginning is that Black's Queen is on c8 not f5. Also note that 70. Qd4+ Kb1 71. Kh6 Qe6! makes White have to start all over again. But here White's King can emerge effectively. As in the first alternate line above, Black avoided 71...Ka1 in order not to get crosschecked on a1 or drop d4 with check, but finds the King on a2 sensitive again. Moving to the c-file seems to lose quickly as in our present game---see below.
72. Kf7! Qf5+!? 73. Ke7 _ Qe5+
Now 74. Qe6!? Qc7+ leads to a desperate battle of "Tablebase Chess" against the Pin Battery that I think Black survives: 75. Ke8 Qb8+ 76. Kd7 Qb7+ 77. Kd6 Qb8+! 78. Kc5! Qf8+! 79. Kc6! Qa8+! 80. Kb5 Qb8+! In all cases Black has played only-moves to avoid EGTB losses---e.g. it is curious that 79...Qg7 80. Kxd5! wins while it didn't win the previous move with Black's Queen on f8! (Someone please double-check all this...) Now 81. Qb6 seems not so clear: 81...Qe8+ 82. Ka5 Qe1+ 83. Qb4 Qe5! looks like nothing special for White, and ditto 83. Ka6 Qe2+ 84. Ka7 Qe7+ 85. Qb7 Qe3+! 86. Ka8 Qe8+ 87. Qb8 Qc6+ etc. White may have a way of winning in this line, so it certainly needs attention too, but the following seems a more precise and incisive means of reaching the desired goal, if the final position in my mainline is winning for White. Note that 74. Kd7?! gives Black an extra option by 74...Qg7+! 75. Kc8 d4!? 76. Qe6+ Kb2 (White can attempt to force this anyway on 76...Ka1 77. Qe1+ Kb2 78. Qf2+ Ka1 79. Qf1+ and then 80. Qf7; for 76...Ka3!? here see more below) 77. Qf7 Qh8+! 78. Kb7 Qh1+ and it doesn't look clear to me: 79. Kb6 Qh6!? or 80. Ka7 Qg1!? or ...d3!? 81. Qf2+ Kb3! are all obscure. (My MacChess 5.0.1 has examined 41 million nodes after 78...Qh1+ and is currently allowing Black to queen with check in a 4Q position, so don't ask me for any computer evals.:-) White "must" be winning somewhere here, but if not, then White is probably not winning in lines C or D either.
Since 74. Qe6!? Qc7+ 75. Kf8?! Qb8+ 76. Kf7 Qc7+ 76. Qe7 Qf4+ 77. Kg8 Qb8+ 78. Qf8 Qe5 enables White to force the above-marked critical position, Black cannot be better off here with 68...Qc8 than with 68...Qe5+(!), but the resulting positions are still relevant to White's general strategy of running Q-side with his King.
73. Ke7 _
[Addendum 11/8: After posting a query about 74. Qe6 on both BBS-es today I have found a new idea for White: not 81. Qb6 but 81. Ka5!? with the funny idea of Qa6 coaxing Black into ...Qg7 and then a discovered-check combination to force a trade of Queens! The main ideas are shown in these two lines so far: 81. Ka5 Qa7+ 82. Qa6 Qg7 83. Kb4+! (+/-) Kb2 84. Qe2+ Kc1 85. Qe3+ Kc2 86. Qb3+ Kd2 (all forced to avoid Q-trade), but 87. Qxd5+ Kc1 88. Qd6!! is EGTB+-. (A general theme that I first noticed in my "Chess Poetry" post on 52...Kc1 is that White Qd6 can be a death-grip if White's King is active and Black's Queen is not, even though White's pawn is still on g6, if Black's King is not tucked all the way into the corner.) Or 81. Ka5 Qa7+ 82. Qa6 Qc5+ 83. Ka4 (hey, Black almost gave mate:-) Qd4+ 84. Kb5+ Kb1/b2 85. Qb6! looks close to winning. What Black can do is 81...Qd8+! to forcibly displace White's Queen after 82. Qb6 Qg5!, but then 83. Ka6! still prevents the liberating ...d4 owing to 84. Qa5+!, and Black is still in trouble...]