Foley–Hoàng–Merkel's twinning conjecture for $(3+1)$-free posets

Let $P$ be a $(3+1)$-free poset, let $v\in P$, and let $P_v$ be the poset obtained by adding an element $v'$ that is incomparable to $w$ if and only if either $w=v$ or $w$ is incomparable to $v$. Let $\operatorname{Inc}(P)$ denote the incomparability graph of $P$.

Foley–Hoàng–Merkel's twinning conjecture. If $P$ is $(3+1)$-free and $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive, then $X_{\operatorname{Inc}(P_v)}(\mathbf{x})$ is $e$-positive for any $v\in P$.

This is a weakened version of the Stanley–Stembridge conjecture, motivated by the fact that twinning preserves the property of being $(3+1)$-free. The source does not state that this conjecture has been resolved.