Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i(E, F)$ is a non-zero class.

**Question:** Do there always exist integers $n, d$ such that the map $H^n(X, E\otimes\mathcal{O}_X(d))\to H^{n+i}(X, F\otimes \mathcal{O}_X(d))$ induced by $c$ is non-zero?

As Will Sawin notes in the comments below, the answer to the question as stated is no. I would still be very interested in a positive answer to a modified question:

**Question 2:** Do there exist a degree $n\in\mathbb{N}$ and a line bundle $L$ such that the induced map $H^n(X, E\otimes L)\to H^{n+i}(X, F\otimes L)$ is non-zero?

I've tried to construct a class in some $H^n(X, E\otimes \mathcal{O}_X(d))$ that is not killed by $c$ by resolving $E$ and $F$ by direct sums of powers of $\mathcal{O}_X(1)$ but that didn't seem to help. Another thing to note is that, for $i>0$, there is only a finite range of degrees $d$ where both groups $H^n(X, E(d))$ and $H^{n+i}(X,F(d))$ are non-zero for some $n$, because of Serre duality and Serre vanishing.