4 problems

Regularity conjecture. If $H$ is finite-dimensional and cosemisimple, then $H$ is regular.

Classification conjecture. Every finite-dimensional simple partial $H$-comodule is of the form

Simplicity conjecture. The partial $H$-comodule $W\operatorname{\square}^{\bar{H}_e}He$ is simple.

ABV2 conjecture. If $H$ is finite-dimensional and semisimple, then $H_{par}$ is finite-dimensional and semisimple.