equivalent subprojection in $\Bbb M_m(A)$

Question

Let $A$ be a $C^*$ algebra,$p$ is a projection in $\Bbb M_n(A)$,$q$ is a projection in $\Bbb M_m(A)$.If there exists a projection $p_0$ in $\Bbb M_k(A)$ such that $q$ is equivalent to $p\oplus p_0$,does there exists a projection $q_0\in \Bbb M_m(A)$ such that $q_0\leq q$ and $p$ is equivalent to the subprojection $q_0$?