$\mathfrak{sl}_2$-Harish-Chandra modules for $\mathfrak{sl}_2 \ltimes L(4)$

We use analogues of Enright's and Arkhipov's functors to determine the quiver and relations for a category of $\mathfrak{sl}_2 \ltimes L(4)$-modules which are locally finite (and with finite multiplicities) over $\mathfrak{sl}_2$. We also outline serious obstacles to extend our result to $\mathfrak{sl}_2 \ltimes L(k)$, for $k>4$.