Pointedness conjecture for non-semisimple Hopf algebras of dimension $pq^2$

Let $\Bbbk$ be an algebraically closed field of characteristic zero. Let $p$ and $q$ be distinct primes, and let $H$ be a non-semisimple Hopf algebra over $\Bbbk$ of dimension $pq^2$. Write $H^*$ for its dual Hopf algebra, and call a Hopf algebra pointed when all its simple comodules are one-dimensional.

Pointedness conjecture. The Hopf algebra $H$ or its dual $H^*$ is pointed.

This conjecture is motivated by the classification results for non-semisimple Hopf algebras of dimension $4p$, including the cases $p\leq 11$, together with related results cited in the source. The general assertion for dimension $pq^2$ remains open in the supplied text.