This is a follow up on an earlier question.

Let $A$ be a non-separable unital (not necessarily self-adjoint) operator algebra, which is reflexive as a Banach space. Let $W$ be a $1$-complemented separable unital subalgebra.

Let's say that $W$ has property (#) if there exist a contractive projection $P:A\to A$ onto $W$ satisfying $P(ux) = uP(x)$ for all $u\in W$, $x\in A$.

Question: Does $W$ have property (#) ? If not, does there exist another $1$-complemented separable subalgebra with this property, which contains $W$?

PS: Could you please share general references in the comments about projections on Banach algebras, specifically about contractive projections whose range is a subalgebra?