Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [Lindenstrauss,1966]. It is straightforward to show that every separable subalgebra is contained in a separable subalgebra $W$ of the form $$W=\bigcup_{n=1}^{\infty} E_n$$ where $(E_n)$ is an increasing sequence of separable 1-complemented subspaces of $A$.

Question: Is it true that every separable subalgebra of $A$ is contained in a separable complemented subalgebra?