Unique real critical point conjecture for the aED problem

Let $U$ be a nonnegative tensor whose entries sum to $1$, and let $\mathcal{M}_\Sigma$ be a marginal independence model contained in the probability simplex $\Delta_{D-1}$. Let the aED problem denote the Euclidean distance optimization problem for this model. Unique real critical point conjecture. The aED problem has precisely one real critical point, namely the point $\hat P \in \mathcal{M}_\Sigma \cap \Delta_{D-1}$ that is closest to $U$. The conjecture is motivated by computational experiments for marginal independence models; the source gives no resolution, so its general validity remains open.