Minimum 2-vertex strongly biconnected spanning directed subgraph problem

A directed graph $G=(V,E)$ is strongly biconnected if $G$ connected and its underlying biconnected. called $2$-vertex-strongly $|V|\geq 3$ the induced subgraph on $V\setminus\left\lbrace w\right\rbrace $ for every vertex $w\in V$. In this paper we study following problem. Given a $G=(V,E)$, compute an edge subset $E^{2sb} \subseteq E$ of minimum size such that $(V,E^{2sb})$