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

$\mathcal{N} = (2,2)$ extended $\mathfrak{sl}(3|2)$ Chern-Simons $AdS_3$ supergravity with new boundaries