A manifold $X$ has the fixed-point property if for every continuous map $f:X→X$ there is $x∈X$ with $f(x)=x$. Examples of such spaces are disks and the real projective plane $\mathbb{RP}^2$.

**Question:** If a compact manifold $X$ has the fixed-point property, does $X\times X$ necessarily have the fixed-point property?

**Known:**

False for $X$ a polyhedron. See An example in the fixed point theory of polyhedra. Bull. Amer. Math. Soc. 73(1967), 922-924.

False for various other spaces. See MO question 283930.

True for compact surfaces: the only such surface with the fixed-point property is $\mathbb{RP}^2$, and $\mathbb{RP}^2\times \mathbb{RP}^2$ also has the fixed-point property, by the Lefschetz fixed point lemma.