Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\sum_i a_{ii}\Bigr\rVert?$$

Attempt: Choose a faithful representation $A \subseteq B(H)$. Then $M_n(A) \subseteq M_n(B(H)) = B(H^{n})$ so we have $$\lVert(a_{ij})\rVert = \sup_{(x_1, \dotsc, x_n) \in (H^n)_1} \sum_j \Bigl\lVert\sum_i a_{ji} x_i\Bigr\rVert.$$

How to proceed?