On the motive of the nested Quot scheme of points on a curve

Let $C$ be a smooth curve over an algebraically closed field $\mathbf{k}$, and let $E$ locally free sheaf of rank $r$. We compute, for every $d>0$, the generating function motives $[\mathrm{Quot}_C(E,\boldsymbol{n} )] \in K_0(\mathrm{Var}_{\mathbf{k}})$, varying $\boldsymbol{n} = (0\leq n_1\leq\cdots\leq n_d)$, where $\mathrm{Quot}_C(E,\boldsymbol{n} )$ is nested Quot scheme points, parametrising $0$-dimensional subsequent quotients $E \twoheadrightarrow T_d \cdots T_1$ fixed length $n_i \chi(T_i)$. The resulting series, obtained by exploiting Bialynicki-Birula decomposition, factors into product shifted motivic zeta functions $C$. In particular, it rational function.