Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$.

If for every subsheaf $\mathcal F \subset \mathcal E$ the following inequality holds $$ \frac{c_1(\mathcal F)\cdot H^{n-1}}{\mathrm{rank}(\mathcal F)} \le \frac{c_1(\mathcal E)\cdot H^{n-1}}{\mathrm{rank}(\mathcal E)} $$ then we say that $\mathcal E$ is semistable.

Mehta-Ramanathan proved that if $\mathcal E$ is semi-stable then its restriction to a sufficiently general curve $C$ cut out by elements of $|mH|$, $m \gg 0$, is also semistable.

Question.If we further assume that $X$ is rationally connected can we ensure the existence of a very free morphism $i: \mathbb P^1 \to X$ such that $i^*\mathcal E$ is semistable?

Recall that $i: \mathbb P^1 \to X$ is a very free morphism if and only if $i^* TX$ is ample.

As observed by mdeland in the comments, $i^* \mathcal E$ is semistable if and only if $$i^* \mathcal E \simeq \mathcal O_{\mathbb P^1}(k)^{\oplus \mathrm{rank}(\mathcal E)} .$$ In particular $\deg(i^* \mathcal E)$ is divisible by $\mathrm{rank}(\mathcal E)$.

If we start with an arbitrary very free morphism $i: \mathbb P^1 \to X$ we can compose it with a rational map $\varphi : \mathbb P^1 \to \mathbb P^1$ of degree $m \cdot \mathrm{rank}(\mathcal E), m \in \mathbb N$, to obtain a new morphism $i' = i \circ \varphi$. The original question seems to be related to the problem.

Problem.Describe the splitting type of a general deformation of $i' > : \mathbb P^1 \to X$ for $m \gg 0$.

Any ideas/references on how to tackle this kind of problem ?