Hecke curves in Frobenius strata of moduli space of rank 2 vector bundles

Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g⩾2$ over $k$. Let ${\mathcal{M}}_X^s(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector bundles with determinant $\mathcal{L}$ on $X$. The Frobenius stratification measures the instability of bundles in ${\mathcal{M}}_X^s(r,\mathcal{L})$ under pullback by the Frobenius map. We show that there exists a Frobenius stratum in ${\mathcal{M}}_X^s(2,\mathcal{L})$ which is covered by Hecke curves.