Polynomial degree conjecture for secant varieties of Gaussian moment varieties

Let $\mathcal{G}_{n,d}$ be the Gaussian moment variety of order $d$ in $n$ variables, and let $\operatorname{Sec}_k(\mathcal{G}_{n,d})$ denote its $k$-th secant variety. Fix $k$ and $n$, and consider the degree of this secant variety as a function of the moment order $d$. Polynomial degree conjecture. For fixed $k$ and $n$, the function

$$d\longmapsto\deg\operatorname{Sec}_k(\mathcal{G}_{n,d})$$

is a polynomial in $d$, starting from the smallest value of $d$ for which the secant variety does not fill the ambient space. The preceding discussion emphasizes that even determining defining ideals can be difficult, so this conjecture concerns a more accessible numerical invariant; its general validity remains open.