Twin-free reduction conjecture for $(3+1)$-free posets

Let $P$ be a $(3+1)$-free poset, meaning that it contains no induced subposet isomorphic to the disjoint union of a $3$-chain and a $1$-chain. Call $P$ twin-free when there is no graph $G$ with a vertex $v$ such that $\operatorname{Inc}(P)=G_v$, where $G_v$ is obtained by twinning $G$ at $v$.

Twin-free reduction conjecture. If $P$ is $(3+1)$-free and twin-free, then $X_{\operatorname{Inc}(P)}(\mathbf{x})$ is $e$-positive.

The source says that, if the preceding $(3+1)$-free twinning conjecture is solved, the Stanley–Stembridge conjecture can be reduced to this conjecture. Its independent resolution status is not given.