oa.operator algebras – Is the unit ball of $A odot B$ strictly dense in $M(A otimes B)$?

Let $A$ and $B$ be $C^*$-algebras and let $A otimes B$ their minimal tensor product and $M(A otimes B)$ the associated multiplier algebra.

On $M(A otimes B)$, we consider the strict topology which is the locally convex topology generated by the seminorms
$$M(Aotimes B)ni xmapsto |x c| $$
$$M(A otimes B)ni xmapsto |cx| $$
for all $c in Aotimes B$. This topology is weaker than the norm-topology on $M(A otimes B)$.

It is easy to prove that the algebraic tensor product $A odot B$ is strictly dense in $M(A otimes B)$: simply observe that $A odot B$ is norm-dense in $A otimes B$ and $A otimes B$ is strictly dense in $M(A otimes B)$.

Question: Is the unit ball of $A odot B$ dense in the unit ball of $M(A otimes B)$?