Solution

This chess riddle was rather complex full of side variations, so I will restrict myself to show you the most beautiful variations which illustrate the concepts.

Before giving you the solution I congratulate Hive account@juanmi96 for the win in this competition and his amazing efforts to solve the problem. I will also check (and upvote) other good attempts!

As some of you found out, the first white move is
1. Kg3!
(The idea is to attack the black rook on a8, and if black takes the white queen, then the white bishop can give a check from b3.)

Now there are several reasonable responses for black:

a)

1. ... Qxh1
2. Bb3+ Kh8 (Qd5 doesn't change anything)
3. Ng6+! hxg6
4. Rh4# (checkmate)

b)

1. ... Re8
2. Qd5+ (2. Bxe8? loses because of Qxh1) Kh8
3. Nf7+ Kg8
4. Nh6++ (double check) Kh8
5. Qg8+! Rxg8
6. Nf7# (smothered mate)

c)

1. ... Rd8 (to prevent Qd5+, but ...)
2. Qd5+! Rxd5
3. a8R+ (and checkmate in two moves)

d)

1. ... h5 (to create an exit path for the king)
2. Qxa8+ Kh7
3. Qf3! (threatens Qxh5+, then a8R; the fastest way) Qg1+
4. Kh3

And now ...

d_x):

4. ... Qaf1+
5. Qxf1 Qxf1+
6. Rxf1 Bxe5
7. Bc2+ g6
8. Rf7+ Kg8
9. Bxg6 (and the a8R followed by checkmate)

d_y):

4. ...Qxg5
5. Bc2+ g6
6. Qb7+ Kh6
7. Nf7+ Kg7
8. Nxg5+ Kh6
9. Qh7+ Kxg5
10. Qxg6+ Kxf4
11. Qg3# (checkmate)

e)

1. ... Qg1+
2. Kh3! Qxh2+ (wants to deflect the white queen from a8)
3. Qxh2

And now ...

e_x):

3. ... Bxe5
4. Qg2! (back to the diagonal) h5 (Re8, then a8R)
5. g6! Rc8
6. a8R Qc3 (other moves are hopeless, too)
7. Bb3+ Kh8
8. Rf8+ Rxf8
9. Rxf8# (checkmate)

e_y):

3. ... Qxa4
4. Qg2 (back to the diagonal) Rd8
5. Rxd4! Qxd4
6. Qa2+ Kh8
7. Nf7+ Kg8
8. Nh6++ Kh8
9. Qg8+ Rxg8
10. Nf7# (smothered mate again)

e_z):

3. ... Qa3
4. Qe2! (fastest) Bxe5
5. Qf3! Rc8
6. a8Q Bb8
7. Qac6 Rxc6
8. Bb3+ Qxb3 (other moves only extend the game)
9. Rf8# (checkmate)

f)

1. ... h6
2. Qxa8+ Kh7
3. g6+ Qxg6
4. Nxg6 Qg1+
5. Kh3 (fastest) Qxg6
6.Qd5! Be5 (let's choose the toughest defense)
7. Bc2! Qxc2
8. a8Q h5
9. Qag8+ Kg6
10. Rf6+ Kxf6
11. Qgf7+ Kg5
12. Qxg7+ Kf4
13. Qgxe5# (checkmate; that was the longest of all variations)