Greedy matching conjecture for torus braids

Let $m,n$ be integers, let $T(m,n)$ be the $m$-strand torus braid represented by $(\sigma_1\dots\sigma_{m-1})^n$, and let $M_{\operatorname{gr}}$ be the greedy matching on the graph of the delooped Khovanov complex.

Greedy matching conjecture. The greedy matching is a Morse matching for all torus braids, that is, the graphs

$$G(\Psi\llbracket (\sigma_1\dots\sigma_{m-1})^n\rrbracket,M_{\operatorname{gr}})$$

contain no directed cycles for all $n,m$.

The greedy matching is known to be acyclic for the 2-, 3-, and 4-strand torus braids discussed in the paper, but its acyclicity for arbitrary numbers of strands remains open.