Irreducibility and singularities of some nested Quot schemes

Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g⩾1$. Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$. Given integers $k_1,k_2,d_1,d_2$ such that $r>k_1>k_2>0$, let ${\mathcal{Q}}_{d_1,d_2}^{k_1,k_2}(E)$ denote the nested Quot scheme which parametrizes pair of quotients $[E\twoheadrightarrow F_1\twoheadrightarrow F_2]$ such that $F_i$ has rank $k_i$ and degree $d_i$. We show that these nested Quot schemes are integral, local complete intersection schemes when $d_1\gg d_2\gg 0$ or $d_2\gg d_1\gg 0$.