Let $M$ be a separable von Neumann algebra and let $A$ be a (von Neumann-)dense *-subalgebra. Suppose that $\alpha,\alpha_1,\alpha_2,\dots$ are automorphisms of $M$, such that for every $a \in A$, $$ \alpha_n(a) = \alpha(a) $$ for all $n$ sufficiently large. Does it follow that $\alpha_n$ converges to $\alpha$ in the $u$-topology?

(Also: what about if we weaken the hypothesis to just assuming that $\alpha_n(a)$ converges in norm to $\alpha(a)$, for all $a \in A$?)

This question is inspired by a related MO question, where an explicit example was requested of a sequence of inner automorphisms on the hyperfinite $III_1$- (or $II_1$-) factor which converge to an outer automorphism. An answer involved a sequence of inner automorphisms satisfying, in particular, the hypotheses of my question, although the proof of convergence uses more information. Perhaps other examples answering that question would be available if the answer to my question is yes.