Existence of positive tensor rank decompositions for unital symmetric real tensors

Let $P_{abc}$ be a unital symmetric real tensor, meaning that its indices are symmetric and its unit conditions are $P_{1ab}=P_{a1b}=P_{ab1}=\delta_{ab}$. A positive tensor rank decomposition is a representation

$$P_{abc}=\sum_{i=1}^R \beta_i p_a^i p_b^i p_c^i$$

with positive coefficients and real vectors $p^i$. Existence conjecture. Every unital symmetric real tensor has a positive tensor rank decomposition. Numerical experiments for $N=3,4,5$ found such decompositions for 1,000 random tensors, but no proof is given, so the general assertion remains open.