By Corollary 2.22 in On fusion categories (by Pavel Etingof, Dmitri Nikshych and Viktor Ostrik) any fusion category is equivalent to the category of finite dimensional representations of a semisimple weak Hopf algebra.
Two fusion categories are called Grothendieck equivalent if their Grothendieck rings are equivalent. Two semisimple weak Hopf algebras are Grothendieck equivalent if their fusion categories are so.
Question: Is there a semisimple Hopf algebra Grothendieck equivalent to a strictly weak one?
(we are specifically interested in the finite dimensional complex case)
IOW: Does the fusion ring distinguish the non-weak and the strictly weak cases?
Note that there is a sufficient condition for a fusion ring to be in the strictly weak case: it is the existence of a basic element with a non-integer Perron-Frobenius dimension. So the non-existence of such non-integer Perron-Frobenius dimension is a necessary condition for the non-weak case.