We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in Dualizable Tensor Categories (joint with Christopher Douglas and Chris Schommer-Pries). Note that (for C semisimple) separability is equivalent to semisimplicity Z(C) (see Corollary 3.5.9.), so Theorem 3.6.7 does address exactly this question.

We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in Dualizable Tensor Categories (joint with Christopher Douglas and Chris Schommer-Pries). Note that (for C semisimple) separability is equivalent to semisimplicity of Z(C) (see Corollary 3.5.9.), so Theorem 3.6.7 does address exactly this question.

We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in Dualizable Tensor Categories (joint with Christopher Douglas and Chris Schommer-Pries). Note that separability equivalent to semisimplicity of C and Z(C) (see Corollary 3.5.9.).

We eventually sorted this out, and it appears as (one direction of) Theorem 3.6.7. in Dualizable Tensor Categories (joint with Christopher Douglas and Chris Schommer-Pries). Note that (for C semisimple) separability is equivalent to semisimplicity Z(C) (see Corollary 3.5.9.), so Theorem 3.6.7 does address exactly this question.