Exterior-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 $\operatorname{Quot}_{\mathbb P^1}(\mathbb C^N,n,r)$.

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

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

This predicts the exterior-power Euler characteristics in the higher-rank setting, where numerical $K$-theoretic invariants of Schur functors are largely unexplored; it is presented as a conjectural extension of the established rank-zero formulas.