Separate polynomiality of symmetric-power Chern coefficients

Let $A_{n,d}:=\{\alpha\in\mathbb Z_{\ge0}^n:|\alpha|=d\}$ and $y_\alpha:=\alpha_1x_1+\cdots+\alpha_nx_n$. Define the Chern classes by

$$\prod_{\alpha\in A_{n,d}}(1+y_\alpha)=\sum_{k\ge0}c_k(n,d).$$

For each partition $\lambda\vdash k$, write

$$c_k(n,d)=\sum_{\lambda\vdash k}f_\lambda(n,d)e_\lambda.$$

Separate polynomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is polynomial in $d$ for fixed $n$, and polynomial in $n$ for fixed $d$.

This conjecture asserts separate polynomial dependence on the rank and degree parameters, a stronger structural property than merely having a closed formula. The source gives no resolution status.