What is the highest number of forced checkmates in a legal position?

Question

Having been inspired by a similar question, what legal positions have the most forced checkmates for one side and for both sides?

Answer 1

The answer to this has already been given as an answer in a related question. That answer was 55.

[Title "Ludwig Zagler, Feenschach 1972"]
[FEN "1rB1Q1Q1/1PRQ3Q/2b2k2/2pQ1N1Q/4KRPp/5N2/1R6/B3nn1b w - - 0 1"]

The position is taken from Tim Krabbe's website. According to Krabbe:

There are 12 mates by Rb2; 1 by g5; 1 by Qd5; 7 by d7; 7 by Qe8, 7 by Qg8; 6 by Qh5; 6 by Qh7; and 8 by Nf5, for a total of 55.

The record with both sides to move is 39 according to this Matplus.net thread.

[Title "Ludwig Zagler, Fairy Chess 6/1974"]
[FEN "6B1/2p2R1B/2P1k1NR/2P1q1PP/1pp1Q1p1/brn1K1p1/1b1r2P1/n1b5 w - - 0 1"]