Non-defectiveness conjecture for inverse Gaussian and gamma moment varieties

Let $\mathcal{M}^\mathrm{IG}_d$ and $\mathcal{M}^{\Gamma}_d$ denote the moment varieties associated with the inverse Gaussian and gamma distributions, respectively. For $k\geq 2$, let $\operatorname{Sec}_k(\mathcal{M})$ denote the $k$th secant variety of a variety $\mathcal{M}$. Non-defectiveness conjecture.

$$\operatorname{Sec}_k(\mathcal{M}^\mathrm{IG}_d) \text{ and } \operatorname{Sec}_k(\mathcal{M}^{\Gamma}_d) \text{ are nondefective for all } d\geq 2 \text{ and } k\geq 2.$$

Computations in exact arithmetic verify non-defectiveness for all $d,k\leq 100$. The conjecture asserts that this computationally observed behavior holds for all indicated values of $d$ and $k$.