# Are there separable states $\rho$ with separable pure decompositions requiring $\operatorname{rank}(\rho)^2$ components?

This question is tightly related to What separable $\rho$ only admit separable pure decompositions with more than $\mathrm{rank}(\rho)$ terms?. In there, examples were given of separable states $$\rho$$ with separable decompositions requiring more than $$\operatorname{rank}(\rho)$$ components. In particular, symmetric Werner states have rank $$\binom{n+1}{2}$$ but require no less than $$n^2$$ rank-one separable states to be written.

On the other hand, the standard proof with Carathéodory's theorem sets $$\operatorname{rank}(\rho)^2$$ as the general upper bound for the sufficient number of elements in a separable pure decomposition of $$\rho$$.

Is there any example of a state that saturates this upper bound? In other words, a separable state $$\rho$$ that cannot be decomposed with less than $$\operatorname{rank}(\rho)^2$$ pure product states?