Brower's fixed-point conjecture for definable maps of the square

Let $\mathbb{K}$ be a definably complete expansion of an ordered field, and let $f:[0,1]^2\to[0,1]^2$ be a definable continuous function. Then Brower's fixed-point conjecture. There exists $c\in[0,1]^2$ such that

$$f(c)=c.$$

This is posed as an open problem because transfer from the real case to definably complete structures is not automatic; the paper places it among other problems and counterexamples.