Stable Cellularity in Hopfological Algebra

Let $H$ be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with $\ell=\max\{d:H_d\ne0\}$, and let $A\ge0$ be a locally finite $H$-module algebra with finite-dimensional split semisimple, $H$-trivial $A_0$. Under the gap $A_d=0$ for $0<d<\ell$, the shifted standard cells