Sharpness of torus detection for invertibility of 2-dimensional topological field theories

Let $\mathcal C$ be a symmetric monoidal bicategory, and let

$$\mathcal Z:\mathrm{Bord}_2^\text{fr}\to\mathcal C$$

be a 2-framed 2-dimensional topological field theory. Write $(S^1,\theta_1)$ for the 2-framed circle with framing $\theta_1$.

Sharpness conjecture. There exists a symmetric monoidal bicategory $\mathcal C$ and a 2-framed 2-dimensional topological field theory

$$\mathcal Z:\mathrm{Bord}_2^\text{fr}\to\mathcal C$$

such that $\mathcal Z(S^1,\theta_1)$ is invertible, but $\mathcal Z$ is not invertible.

The preceding result shows that invertibility on the positively framed circle forces invertibility for theories with a spherophilic tangential structure. This conjecture asserts that the corresponding statement fails for general 2-framed theories, so the spherophilic hypothesis would be essential.