Small-polarization stability conjecture for exceptional bundles on Hirzebruch surfaces

Let $e\geq 2$ and let $\mathcal{V}$ be an exceptional bundle on the Hirzebruch surface $\mathbb{F}_e$. Small-polarization stability conjecture. The bundle $\mathcal{V}$ is $\mu_{H_{\epsilon}}$-stable for every sufficiently small $\epsilon>0$. If true, exceptional bundles on $\mathbb{F}_e$ could be determined from the stability intervals of exceptional bundles on $\mathbb{F}_0$ and $\mathbb{F}_1$.