Sturmfels's binomial conjecture for the maximum number of Gaussian-mixture modes

Let $m(d,k)$ denote the maximum possible number of modes of a mixture of $k$ Gaussian distributions in dimension $d$.

Sturmfels's conjecture. For all integers $d,k\geq 1$, the maximum number of modes is

$$m(d,k)=\binom{d+k-1}{d}.$$

This conjecture was proposed in the context of the maximum-number-of-modes problem after the known result $m(d,2)=d+1$. The source does not provide evidence here that the conjecture has been resolved, so its status is recorded as open.