Sufficient conditions for a map $\phi: M(A) \to M(B)$ to have strictly closed image

Question

Let $A$ and $B$ be $C^*$-algebras with multiplier algebras $M(A)$ and $M(B)$. Are there any nice conditions that ensure that a strict (= norm-continuous + strictly continuous on bounded subsets of $M(A)$) linear map $\phi: M(A) \to M(B)$ has strictly closed image. In particular, I'm interested in the following situations:

(1) $\phi$ is a strict $*$-morphism.

(2) $\phi$ arises as the unique extension of a strict linear map $A \to M(B)$.

(3) $\phi$ arises as a unique extension of a strict injective linear map $A \hookrightarrow M(B)$.