The quotient-manifold criterion for Banach–Lie subgroups

Let $G$ be a Banach–Lie group and let $H$ be a closed subgroup. Let $q:G\to G/H$ be the quotient map, $q(g)=gH$.

Quotient-manifold criterion. $H$ is a Banach–Lie subgroup if and only if $G/H$ carries the structure of a Banach manifold for which $q$ has a surjective differential at each point and the action of $G$ on $G/H$ is smooth.

This is a proposed analogue, for general subgroups, of the preceding quotient criterion for closed normal subgroups. The supplied text does not state whether the criterion is known or remains open.