Fulger–Murayama's characterization of projective space by tangent-sheaf Seshadri constants

Let $X$ be a smooth projective variety over an algebraically closed field. For a coherent sheaf $E$ on $X$ and a point $p\in X$, write $\varepsilon(E,p)$ for its Seshadri constant at $p$. Fulger–Murayama's question. If there exists $p\in X$ such that

$$\varepsilon(T_X,p)>0,$$

then $X\simeq \mathbb P^n$. This asks whether positivity of the tangent-sheaf Seshadri constant at one point characterizes projective space; the question is posed in analogy with known characterizations using the anticanonical divisor, and its resolution is not given here.