Asymptotic semistability conjecture for syzygy bundles of Ulrich bundles

Let $X$ be a smooth irreducible projective variety with a fixed ample line bundle $\mathcal{O}_X(1)$, and let $\mathcal{E}$ be an Ulrich bundle on $X$. For an integer $m$, write $\mathcal{E}(m):=\mathcal{E}\otimes\mathcal{O}_X(m)$, and let $S_{\mathcal{E}(m)}$ denote the kernel of the evaluation map

$$H^0(X,\mathcal{E}(m))\otimes\mathcal{O}_X\longrightarrow\mathcal{E}(m).$$

Asymptotic semistability conjecture. There exists an integer $m_0\gg0$ such that, for every $m\ge m_0$, the syzygy bundle $S_{\mathcal{E}(m)}$ is semistable. This is proposed as a guess based on the paper's results and examples: the assertion is not established in the source and is intended to address eventual semistability after sufficiently positive twisting.