fixed points of conjugation by family of matrices

Question

Take a set of $N$ complex-valued $D\times D$ matrices $\{M_i\}$, indexed by $i=1,\ldots,N$, that are linearly independent in the matrix algebra $M_D\mathbb{C}$. Is it true that $N=D^2$, i.e.$$\text{span}(\{M_i\})=M_D\mathbb{C},$$if and only if the solutions $X$ to the equation $$X\sum_i^NM_iM_i^\dagger=\sum_i^NM_iXM_i^\dagger,$$are all proportional to the identity?