Asymptotic Riemann–Roch conjecture for the relative second Chern class

Assume that the base field $k$ has characteristic zero. Let $f:\tilde X\to X$ be the morphism under consideration, and let $\mathcal F$ be a rank $r$ vector bundle on $\tilde X$. Write $\chi(f,-)$ for the relative Euler characteristic and $c_1(f,\mathcal F)$ and $c_2(f,\mathcal F)$ for the relative first and second Chern classes. Asymptotic Riemann–Roch conjecture. One expects

$$\chi(f,\operatorname{Sym}^m\mathcal F)=-\frac{m^{r+1}}{r!}\left(c_1(f,\mathcal F)^2-c_2(f,\mathcal F)\right)+O(m^r).$$

This conjecturally characterizes the relative second Chern class in characteristic zero and is needed in arbitrary rank; the rank-two case is attributed in the source to work of Wang, with the arbitrary-rank version suggested by Langer. The source does not provide evidence of a resolution.