Algebraic identifiability of mixtures of bivariate Gaussians

Let $\mathcal{G}_{n,d}$ denote the Gaussian moment variety of order $d$ in $n$ variables, and let $\operatorname{Sec}_k(\mathcal{G}_{n,d})$ be its $k$-th secant variety. Algebraic identifiability means that a general point of the secant variety has only finitely many decompositions as a sum of $k$ points of the moment variety. Bivariate Gaussian identifiability conjecture. Equality holds in the secant-dimension formula for $n=2$ and all values of $d$ and $k$. In particular, all moment varieties of mixtures of bivariate Gaussians are algebraically identifiable. The preceding theorem establishes the analogous assertion for univariate Gaussians and exhibits failure for $n\geq 3$, while the bivariate assertion is supported by computations up to $d=24$.