Tanaka's strict topological-complexity inequality conjecture for finite spaces

Let $X$ be a finite topological space, let $PX$ be its path space, and let $\operatorname{tc}(X)$ denote its topological complexity. Let $\operatorname{cat}(X)$ denote its Lusternik–Schnirelmann category, so that $\operatorname{cat}(\double{X})$ is the category of the product $X\times X$. Tanaka's conjecture. There exists a finite space $X$ such that

$$\operatorname{tc}(X)<\operatorname{cat}(X\times X).$$

The general inequality $\operatorname{tc}(X)\leq\operatorname{cat}(X\times X)$ is known, and this conjecture asks whether it can be strict for some finite space.