  We present the first example of $\mathcal{N}=(2,2)$ formulation for the
extended higher-spin $AdS_3$ supergravity with the most general boundary
conditions as an extension of the $\mathcal{N} =\,(1,1)$ work, discovered
recently by us [1]. Using the method proposed by Grumiller and Riegler, we
construct a consistent class of the most general boundary conditions to extend
it. An important consequence of our method is that, for the loosest set of
boundary conditions it ensures that their asymptotic symmetry algebras consist
of two copies of the $\mathfrak{sl}(3|2)_k$. Moreover, we enjoin some certain
restrictions on the gauge fields for the most general boundary conditions,
leading to the supersymmetric extensions of the Brown and Henneaux boundary
conditions. Based on these results, we finally find out that the asymptotic
symmetry algebras are two copies of the super $\mathcal{W}_3$ algebra for
$\mathcal{N} = (2,2)$ extended higher-spin supergravity theory in $AdS_3$.
