Finite Quot scheme and tautological Chern-class conjecture

Fix $n\geq 1$, let $e_n=(r,-L,\operatorname{ch}_2(e_n))$ be determined by $\chi(e_n,(1,0,-n))=0$, and set $v_n=e_n+(1,0,-n)$. Let $V_n$ be a vector bundle with Chern character $v_n$, and let $\hat E_n$ have Chern character $e_n^*$. Denote by $(V_n^*)^{[n]}$ and $\det\hat E_n^{[n]}$ the corresponding tautological bundles on the Hilbert scheme $S^{[n]}$.

Finite Quot tautological Chern-class conjecture. One has

$$c_{2n}\left((V_n^*)^{[n]}\right)=\chi\left(S^{[n]},\det\hat E_n^{[n]}\right).$$

This equality is motivated by interpreting the top Chern class as the expected count of sections whose induced map drops rank, and by the finite Quot scheme method. The source describes it as an expectation supported by numerical evidence; no proof or disproof is given.