Self-Kronecker-product lower-bound conjecture for nonnegative rank

Let $A$ be a nonnegative matrix, let $A\otimes A$ denote its Kronecker square, and write $\operatorname{rank}(A)$ for its ordinary rank. Self-Kronecker-product conjecture.

$$\operatorname{rank}_+(A\otimes A)\geq \operatorname{rank}_+(A)\operatorname{rank}(A).$$

The source introduces this as a new conjecture motivated by examples concerning Kronecker products; its resolution is not given.