Defect pattern conjecture for fourth-order Gaussian moment varieties

Let $\mathcal{G}_{n,4}$ be the Gaussian moment variety of order $4$ in $n$ variables, and let $\delta_j$ denote the defect of its $j$-th secant variety. The model-parameter count is compared with the ambient dimension $\binom{n+4}{4}-1$. Fourth-order Gaussian defect conjecture. The Gaussian moment variety $\mathcal{G}_{n,4}$ is $(n+3)$-defective with defect $\delta_{n+3}=1$ for $n\geq 8$. Furthermore, for all $r\geq 3$, the $(n+r)$-defect of $\mathcal{G}_{n,4}$ is

$$\delta_{n+r}=\binom{r-1}{2},$$

unless the number of model parameters exceeds the ambient dimension $\binom{n+4}{4}-1$. The pattern is supported by computations verified for $n\leq 14$, but no general proof is supplied.