The split exactness conjecture for matching-complex deletion sequences

Let $n\geq3$, let $e$ be an edge of the complete graph on $n$ vertices, and let $\mathsf{M}_n\setminus e$ be the matching complex with the vertex $e$ deleted. Consider the sequence

$$0\longrightarrow\tilde{H}_d(\mathsf{M}_n\setminus e;\mathbb{Z})\longrightarrow\tilde{H}_d(\mathsf{M}_n;\mathbb{Z})\longrightarrow\langle e\rangle\otimes\tilde{H}_{d-1}(\mathsf{M}_{[n]\setminus e};\mathbb{Z})\longrightarrow0.$$

Split exactness conjecture. This sequence, obtained by cutting the long exact $0$-$e$-$2$ sequence, is split exact for every $n\geq3$ and every $d$.

If true, the long exact deletion sequence would decompose into short split exact sequences, giving a direct-sum description of the homology groups. The supplied text gives no proof or resolution, so the conjecture remains open.