Additivity of nonnegative rank under Cartesian products

Let $P_1$ and $P_2$ be polytopes. Their nonnegative rank is the smallest number of nonnegative factors in a nonnegative factorization of a slack matrix of the polytope. Product additivity conjecture. The nonnegative rank is additive under Cartesian products:

$$\mathrm{rk}_+(P_1 \times P_2)=\mathrm{rk}_+(P_1)+\mathrm{rk}_+(P_2).$$

This conjecture would describe how nonnegative rank behaves under direct sums of matroids, since direct sums correspond to Cartesian products of matroid base polytopes. It was first asked during a Dagstuhl seminar in 2013; the source gives no resolution.