Let $P$ be a minimization primal problem $\min c^T x$. Does $P$ and its dual $P^*$ always have the same number of optimal solutions?

Question

Let $P$ be a minimization primal problem $\min c^T x$ and let $P^*$ be its dual.

I've been wondering about the following:

Suppose $P$ has exactly $n$ optimal solutions.

I know that $P^*$ also has an optimal solution $\bar u$ with the same objective value by Strong Duality.

Does $P^*$ also has exactly $n$ optimal solutions then ?

I know about Complementary Slackness, but haven't yet come across an example where the equality doesn't hold.