Let $A$ and $B$ be $C^*$-algebras. Suppose that there exists a projection $p$ in $\mathcal{M}(B)$, the multiplier algebra of $B$, such that $A=pBp$. That is, $A$ is a corner of $B$.
Question: Is it true that the corner $p(\mathcal{M}(B))p$ contains the multiplier algebra of $A$ (which we view as a subalgebra of $B$)? In other words, does every multiplier of the corner come from the corner of the multiplier algebra?