Projective-dimension characterization of split links

Let $B$ be an $m$-component closed braid, let $H(B)$ denote its HOMFLYPT homology, and let $R_B$ be the associated ring. A link is $n$-split if it is equivalent to a split union of $n$ closed braids.

Split-link characterization. $B$ represents an $n$-split link if and only if

$$\operatorname{pd}_{R_B} H(B) \leq m-n.$$

The preceding theorem proves the forward implication and gives the same upper bound for an $n$-split link, while the conjecture asserts the converse. If true, the projective dimension of HOMFLYPT homology would detect how many times a link can be split, rather than measuring its distance from being split via splitting number.