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Agree that it should be able to capture with a mobility of 1.
OK, I changed the king to work as in orthodox chess. If it has limited mobility, then it should be able to capture.

I should know better than to comment when not feeling well, but have never learned that particular lesson. Sorry about the bede jumping comment. However, the king can reach 25 - counting its starting position - squares. How many pieces would be required to cover all 25 squares, and prevent the king from capturing any of them? How likely are those positions to arise?
I specified that the bede can jump any agent when jumping in the orthogonal directions, and that can jump even if the jumped square is unoccupied. The king is unable to take out any agent, and there are quite a lot of powerful pieces on the board, so I thought it should have some chances of escape. But, yeah, maybe moving through check makes it difficult to mate. I could change it, or perhaps make it an alternative rule?

Two comments: 1) It's not specified if the colorbound bede can ignore pieces on the opposite color, or if those pieces block the bede's move. 2) How checkmatable is the king? It's an extremely powerful piece, and allowing it to move through check on the first step of its 2 step move makes it even more powerful.
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This article is too vague and inconsistent to make a Diagram for it. I have for instance no idea how a Squire moves.