Torsion lower-bound conjecture for homology of finite spaces

Let $X$ be a finite $T_0$-space and let $k\in\mathbb{N}$ with $k\geq 2$. Write $\#X$ for the number of points of $X$, and let $H_k(X)$ denote its integral homology group.

Torsion lower-bound conjecture. If $H_k(X)$ has torsion, then

$$\#X\geq 2k+11.$$

This conjecture extends the paper's small-cardinality torsion result to higher homological degrees. It gives a linear lower bound on the size of a finite model carrying torsion in $k$-th homology; the source states it without indicating that it has been resolved.