Divisorial instability conjecture for blowups of Fano hypersurfaces along linear subspaces

Let $\Gamma \simeq \mathbb P^k \subset X$ be a $k$-dimensional linear subspace in a smooth Fano hypersurface $X \subset \mathbb P^{n+1}$ of degree $d$. Let $\varphi \colon Y \rightarrow X$ be the blowup of $X$ along $\Gamma$, and suppose that $Y$ is a Fano variety. Divisorial instability conjecture. There exists a constant $c(d)$ depending only on $d$ such that

$$k > c(d)$$

if and only if $Y$ is divisorially unstable for all $n \geq k+2$. The preceding theorem verifies a related bound for $k \geq 3\cdot\deg X$ and $k+2 \leq n \leq 1000$, while the conjecture seeks a degree-dependent threshold valid for all $n \geq k+2$.