I think the theoretical minimum is 12.5 moves.
But practically I think 13 is not achievable. Just because you need to capture the queen on the way. And if you do it on move 5 then you need to wait anyway for the light squared bishop to move. Furthermore the king is going for a walk. It is way too specific to happen.
playstrategy.org/study/0iEcxmAo

I do not think 14 is practically possible. You can go from h1 to a1 to a8 but to h8 there is no way.

Entries for Weekly Challenge 172 has closed. Thanks to all users who participated.

The winner is @LowestPosWinRate who won a game of Monster Chess with the King visiting all 4 corners in just 29 turns in the following game: playstrategy.org/hJBWCcfJ

Users are still welcome to attempt this challenge and post answers here, but the winner has been fixed. The next challenge can be found here: playstrategy.org/forum/weekly-challenges/weekly-challenge-173-escaping-oware-capture

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