Dimension conjecture for secant varieties of cubic Gaussian moment varieties

Let $\mathcal{G}_{n,3}$ be the Gaussian moment variety of order $3$ in $n$ variables, and let $\operatorname{Sec}_k(\mathcal{G}_{n,3})$ denote its $k$-th secant variety. For $K$ defined so that $K+1$ is the smallest integer for which the displayed expression exceeds the ambient dimension $\binom{n+3}{3}-1$, dimension conjecture for cubic Gaussian moment varieties. For any $n\geq 2$ and $k\geq 1$, with $k=1,2,\ldots,K$, one has

$$\dim\operatorname{Sec}_k(\mathcal{G}_{n,3})=\frac{1}{6}k\left[k^2-3(n+4)k+3n(n+6)+23\right]-(n+2).$$

This conjecture is based on computations of higher secant varieties and extends the proven low-dimensional results; its general validity remains open.