I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying to assign probabilities $p_1,\dots,p_n$ to these $n$ points in such a way that the mean of the distribution is $\mu$ and the covariance is bounded by $\Sigma$ by a linear matrix inequality, i.e. the following problem: $$\textrm{maximize}_{p_1,\dots,p_n} \sum_i \|x_i\|p_i $$ subject to $$\sum_i p_i = 1$$ $$\sum_i x_i p_i = \mu$$ $$\sum_{i}(x_{i}-\mu)(x_{i}-\mu)^{T}p_{i}\preceq\Sigma $$ $$p_i\geq 0 \forall i$$

I'm solving this numerically and every optimal solution has at most $7$ entries that are nonzero. Is it possible to prove that this must always be the case? If I were to omit the covariance constraint, I'd just have a standard linear program that could be dealt with using complementary slackness conditions, but I have no such luxury here.