Strong binomial-polynomiality conjecture for Chern classes of polynomial spaces

Let $\Pol^d(\mathbb{C}^n)$ denote the relevant polynomial-space bundle, and let $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ be its $k$th Chern class. For integers $a_i,b_i$, set

$$B_i=\binom{d+n+a_i}{n+b_i}.$$

Strong conjecture. The class $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ is of the form $P(B_1,\dots,B_s)$, where $P$ is a polynomial. This strengthens the weak conjecture by proposing an explicit binomial-coefficient form, consistent with the displayed low-degree formulas; no resolution is given in the paper.