Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.

Theorem. Suppose $P$ and $Q$ are posets with the FPP and at least one of them is finite. Then $(P\times Q)$ has FPP.

Note: $(a,b)\le(c,d)$ if and only if $a\le c$ and $b\le d$.

Question. Suppose $P$ and $Q$ are two infinite posets with the FPP. Does $(P\times Q)$ have the FPP ?