Criterion for the existence of a $1$-Lipschitz selection from the metric projection onto the set of continuous selections from a multivalued mapping

Let $S_F$ be the set of continuous bounded selections from the set-valued mapping $F\colon T \rightarrow 2^H$ with nonempty convex closed values; here $T$ is a paracompact Hausdorff topological space, and $H$ is a Hilbert space. In this paper, we obtain a criterion for the existence of a $1$-Lipschitz selection from the metric projection onto the set $S_F$ in $C(T,H)$.