I wonder the state of the following conjecture in "Deformation theory, Kontsevich, Soibelman":
Conjecture 3.3.5. Rigid [abelian symmetric] tensor categories [over an algebraically closed field $k$] with ranks in $\mathbb{Z}$ can be of two types: comodules over commutative Hopf algebras or comodules over supercommutative Hopf algebras.
Deligne's result gives the answer for the case $\mathrm{char}(k)=0$ and nonnegative ranks: comodules over a commutative Hopf algebra.