Wang's universal Chern-class conjecture for tautological bundles on symmetric products of curves

Let $C$ be a smooth projective curve and let $E_r$ be a vector bundle of rank $r$ on $C$. Write $C^{[n]}$ for the Hilbert scheme of $n$ points on $C$, $c(E_r^{[n]})$ for the total Chern class, $d_r=\int_C c(E_r)$, and $e$ for the Euler number of $C$. Let $A_n^r$, $B_n^r$, $C_n^r$, and $D_n^r$ be integers depending only on $r$ and $n$.

Wang's conjecture. One has

$$\sum_{n=0}^{\infty} z^n \int_{C^{[n]}} c(E_r^{[n]})=\exp\left(\sum_{n=1}^{\infty} \frac{z^n}{n}(A_n^r d_r+B_n^r e)\right),$$

and

$$\sum_{n=0}^{\infty} \frac{z^n}{n} \int_{C^{[n]}} c(-E_r^{[n]})=\exp\left(\sum_{n=1}^{\infty} \frac{z^n}{n}(C_n^r(-d_r)+D_n^r e)\right),$$

where

$$A_n^r=(-1)^{n+1}\binom{rn-1}{n-1},\qquad C_n^r=(-1)^n\binom{-rn-1}{n-1}=(-1)^{n-1}A_n^{r+1},\qquad D_n^r=(-1)^nB_n^{r+1}.$$

The conjecture predicts universal coefficients for total Chern-class integrals on symmetric products of curves. The paper establishes the relevant cases and reports verification for several ranks and low values of $n$, but the supplied text does not establish the full conjecture; it also notes that the analogous surface conjecture is known for K3 surfaces.