The free-rank conjecture for HOMFLYPT Yang–Baxter homology

Let $R_{(m)}$ be the unital Yang–Baxter operator giving the HOMFLYPT polynomial on level $m$, and let $H_n(R_{(m)})$ denote its homology. The free part of $H_n(R_{(m)})$ means its direct summand consisting of free $k$-modules.

Free-rank conjecture. The rank of the free part of $H_n(R_{(m)})$ is $2^{m-1}$ for $n\geq m-1$, while

$$\operatorname{rank}(H_n(R_{(m)}))=\binom{m-1}{0}+\binom{m-1}{1}+\cdots+\binom{m-1}{n}$$

for $0<n\leq m-1$.

The source says that the displayed right-hand side was proved to be a lower bound for the rank, but does not state that equality was proved. Thus the full conjecture remains open on the supplied evidence.