We say that a simple, undirected graph $G=(V,E)$ has the fixed point property (FPP) if for every graph homomorphism $f:G\to G$ there is $v\in V$ such that $f(v) = v$.

If $G$ has FPP, does $G\times G$ have FPP (where by $\times$ we denote the Tensor or categorical product of graphs)?