Weak 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. Weak conjecture. For any fixed $k$ and $d$, the class $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ is a polynomial in $n$. The conjecture is supported by explicit formulas for $k=1$ and $k=2$ and by computer calculations. The paper notes that it implies separate polynomiality in $n$ and $d$, but not polynomiality jointly in $n$ and $d$.