The representation theory of the minimal series principal $W$-algebras was well-studied. In particular, Arakawa proved the $C_2$-cofiniteness and the rationality of their modular invariant representations. The other principal $W$-algebras generally lack such good properties. Recently, many people have intensively studied the tensor categories of $W^k(\mathfrak{sl}_2)$-modules for general $k\in \mathbb{C} \setminus \{-2\}$. In this talk, I will report on the recent progress in understanding the tensor category of $V^{13+6p/q+6q/p}(Vir)$-modules based on the joint work with McRae and Yang.