This is a follow up on an earlier question.

In [Lau&Loy, 2008] a Banach algebra $\mathcal{U}$ was called to have the Tomiyama property if any contractive projection $P:\mathcal{U}\to \mathcal{U}$, whose range $P(\mathcal{U})$ is a subalgebra, is a conditional expectation.

Analogously, let's say that $\mathcal{U}$ has the "separable Tomiyama property" if any contractive projection, whose range is a separable subalgebra, is a conditional expectation. Let's also say that $\mathcal{U}$ has property ($T_\omega$) if every $1$-complemented separable subalgebra of $\mathcal{U}$ is contained in a unital separable subalgebra, which is a range of a conditional expectation. Similarly, $\mathcal{U}$ has property left ($T_\omega$) ( resp. right ($T_\omega$) ) if every $1$-complemented separable subalgebra of $\mathcal{U}$ is contained in a unital separable subalgebra $W$, which is a range of a contractive projection $P$ satisfying $P(ux) = uP(x)$ ( resp. $P(xu) = P(x)u$ ) for all $u\in W$, $x\in A$.

Question: Let $A$ be a reflexive, non-separable, simple, unital (not necessarily self-adjoint) operator algebra, which possesses no minimal idempotents, i.e., $socle(A)=\{0\}$. Does $A$ have property ($T_\omega$)?

It would be great if there were an answer for the other properties instead of (or in addition to) property ($T_\omega$).

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