Strong binomiality conjecture for 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

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

Strong binomiality conjecture. For each fixed $k$ and each partition $\lambda\vdash k$, the coefficient $f_\lambda(n,d)$ is a polynomial in finitely many binomial coefficients of the form

$$\binom{d+n+a}{n+b}.$$

This refines separate polynomiality by predicting dependence through a distinguished family of binomial coefficients. The source gives no resolution status.