Minimal-system nonrecurrence under two transformations

A topological dynamical system is a pair $(X,T)$ consisting of a compact space $X$ and a continuous map $T:X\to X$; it is minimal if every orbit is dense. For two transformations $T$ and $S$ on the same space, $(x,x)$ denotes the point in $X\times X$, acted on by $T\times S$. Minimal-system nonrecurrence conjecture. There are minimal systems $(X,T)$ and $(X,S)$ such that $(x,x)$ is not recurrent under $T\times S$ for any $x\in X$. This asks whether simultaneous recurrence can fail everywhere even when both systems are minimal; without minimality, examples are known, while the minimal case is posed as open.