Symmetric-power Euler characteristic conjecture for higher-rank Quot schemes

Let $N$ and $r$ be as in the higher-rank Quot scheme, write $n=(N-r)a+b$ with $0\leq b<N-r$, and let $L\to\mathbb P^1$ be a line bundle. The tautological complex $L^{[n]}$ is defined on $\operatorname{Quot}_{\mathbb P^1}(\mathbb C^N,n,r)$.

Symmetric-power conjecture. For every $k\leq n+r(a+1)$,

$$\chi\left(\operatorname{Quot}_{\mathbb P^1}(\mathbb C^N,n,r),\operatorname{Sym}^kL^{[n]}\right)=\binom{N\chi(L)+k-1}{k}.$$

This is the proposed symmetric-power analogue of the known rank-zero formulas and concerns a higher-rank case whose numerical $K$-theoretic invariants are largely unexplored; its resolution is not given here.