Matsuda–Ozeki–Yamashita's branch-vertex conjecture for claw-free graphs

Let $k$ be a non-negative integer and let $G$ be a connected claw-free graph of order $n$. For an integer $r\geq 2$, let $\sigma_r(G)$ be the minimum degree sum of an independent set of $r$ vertices in $G$:

$$\sigma_r(G)=\min\left\{\sum_{x\in S}\deg_G(x): S\subseteq V(G),\ |S|=r,\ S\text{ is independent}\right\}.$$

A vertex of degree at least three in a tree is called a branch vertex. Matsuda–Ozeki–Yamashita's conjecture. Let $k$ be a non-negative integer and let $G$ be a connected claw-free graph of order $n$. If

$$\sigma_{2k+3}(G)\geq n-2,$$

then $G$ has a spanning tree with at most $k$ branch vertices. This conjecture proposes a sharper degree-sum condition for spanning trees with boundedly many branch vertices in connected claw-free graphs, improving on earlier sufficient conditions. The supplied text does not state whether the conjecture has been resolved.