Let $X$ be a strict (LB)-space, i.e. the locally convex inductive limit of a strict inductive sequence of Banach spaces $X_n$. Then $X$ is linearly homeomorphic to a quotient $(\bigoplus_n X_n) / F$ by a closed subspace $F$ of the locally convex direct sum $\bigoplus_n X_n$. Is $F$ a complemented subspace?
A sufficient condition would be that $F$ has countable codimension. (In a barrelled space any algebraic complement of a countably codimensional subspace is topological, see [Wilansky, "Modern Methods in Topological Vector Spaces", Theorem 13-3-19]).
If this is not true, are there any other known simple conditions on the strict inductive sequence such that $F$ is complemented?