Self-Referential Puzzle II

Here is a redrawing of the second self-referential number square. This focuses around the concept of having an incorrect clue, and so as well as the self-referential part of it you need to be able to prove which clues are true and which are false.

Note that the addition box on the grid is assumed to be true, and the two bold phrases (that one of the clues is false and that it is a Latin Square using the digits 1 to 5) are true as well. After all, it’s a puzzle not a trick.

Last time I posted one of my Self-Referential squares I put the solution in the post. I don’t think that worked so well, so here is the solution as a separate web-page.