Which player has the better chances in the following position?
While having an extra pawn is usually preferable, when the only pieces on the board are the queens, a single pawn that is far advanced can be worth more than several pawns that are not if the advanced pawn cannot be blockaded by the opponent's king. In Patel v. Monsen, the players agreed to a draw after 44...Qg7 45.Qf4+. While a draw was not an unreasonable result, had the match still been in doubt, it would have been reasonable for Black to have played for a win.
The problem for White is that the only way to stop the Black pawn from advancing is by keeping the Black king in check. As a result, he does not have the time he needs to advance his own pawns. The problem for Black is that his king is in the of a wide open board and White has lots and lots of checking opportunities. Nevertheless, White has to careful not to deliver a check that can be blocked with a counter-check or with a pin that will force the exchange of queens. For example, after 44...Ke7, 45.Qe4+ would be met with 45...Qe6 forcing the queens of the board. Black may be able to use this possibility to leave White without a good check for a move or two which is all Black needs to advance the pawn.
One possibility after 44...Ke7 is 45.Qg5+ Qf6 46.Qc5+ Qd6 47.Qg5+ Kf7 48.Qf5+ Qf6 49. Qh7+ Qf8.
White now has no checks and he must try to prevent the pawn from advancing with 50. Qb1. Black then can prepare the advance with 50...Qf7 when Black can force the queens off the board after either 51.Qc8+ Qe8+ or 51. Qc5+ Qe7+. After 51.Kd2 a2 52.Qa1 Qf2+ 53.Kd3 Qxh2, Black has reestablished material equality, but he is going to be subjected to another avalanche of checks. Whether Black can actually force the pawn home is far from clear, nevertheless, there will be plenty of chances for White to go wrong and the winning chances definitely lie with Black.
Two things to remember in endings with queens and pawns on both sides:
(1) Having the most pawns is not as important as having the most advanced pawn.
(2) Even though the board looks wide open, don't assume that perpetual check is inevitable. It is often possible to leave your opponent without any good checks.