Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$.
Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or $H^{*}$, then $\mathcal{C}(S)$ weakly group theoretical.
Suppose also that $H$ admits a nontrivial left coideal subalgebra.

Question: Can we deduce that $\mathcal{C}$ is also weakly group theoretical?

Remark: this question should be weaker than the famous question:
Is a fusion category weakly integral iff it's weakly group theoretical?


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