I think this is a counterexample to the result I was looking for. Let C be the idempotent completion of TL_{-1}, the Temperley-Lieb category with loop value -1. (Or equivalently, the category of finite dimensional representations of U_q(sl_2) where q is an eight root of unity.)

Let B be the Fibonacci category and let B' be its Galois conjugate, let x and x' be the nontrivial objects. Let D_2 be the Deligne tensor product of B and B'. Let F_2 be the functor sending V_2 to $x \boxtimes x'$, this exists by the universal property of Temperley-Lieb (i.e. $x \boxtimes x'$ is symmetrically self-dual and has quantum dimension -1). This functor is dominant because $F_2(V_2 \otimes V_2)$ has every object of D_2 as a summand.

As Victor Ostrik points out F_2 is not a ribbon functor, so this is not a counterexample.

I think this is a counterexample to the result I was looking for. Let C be the idempotent completion of TL_{-1}, the Temperley-Lieb category with loop value -1. (Or equivalently, the category of finite dimensional representations of U_q(sl_2) where q is an third root of unity.)

Let B be the Fibonacci category and let B' be its Galois conjugate, let x and x' be the nontrivial objects. Let D_2 be the Deligne tensor product of B and B'. Let F_2 be the functor sending V_2 to $x \boxtimes x'$, this exists by the universal property of Temperley-Lieb (i.e. $x \boxtimes x'$ is symmetrically self-dual and has quantum dimension -1). This functor is dominant because $F_2(V_2 \otimes V_2)$ has every object of D_2 as a summand.

As Victor Ostrik points out F_2 is not a ribbon functor, so this is not a counterexample.

I think this is a counterexample to the result I was looking for. Let C be the idempotent completion of TL_{-1}, the Temperley-Lieb category with loop value -1. (Or equivalently, the category of finite dimensional representations of U_q(sl_2) where q is an eight root of unity.)

Let D_1 be the quotient of C by negligibles. This category is just Rep(Z/2) but with the dimension of the nontrivial rep being -1.

Let B be the Fibonacci category and let B' be its Galois conjugate, let x and x' be the nontrivial objects. Let D_2 be the Deligne tensor product of B and B'. Let F_2 be the functor sending V_2 to $x \boxtimes x'$, this exists by the universal property of Temperley-Lieb (i.e. $x \boxtimes x'$ is symmetrically self-dual and has quantum dimension -1). This functor is dominant because $F_2(V_2 \otimes V_2)$ has every object of D_2 as a summand.

I think this is a counterexample to the result I was looking for. Let C be the idempotent completion of TL_{-1}, the Temperley-Lieb category with loop value -1. (Or equivalently, the category of finite dimensional representations of U_q(sl_2) where q is an eight root of unity.)

Let B be the Fibonacci category and let B' be its Galois conjugate, let x and x' be the nontrivial objects. Let D_2 be the Deligne tensor product of B and B'. Let F_2 be the functor sending V_2 to $x \boxtimes x'$, this exists by the universal property of Temperley-Lieb (i.e. $x \boxtimes x'$ is symmetrically self-dual and has quantum dimension -1). This functor is dominant because $F_2(V_2 \otimes V_2)$ has every object of D_2 as a summand.

As Victor Ostrik points out F_2 is not a ribbon functor, so this is not a counterexample.