The converse categorification conjecture for the categorical Heisenberg algebra

Let ${\cat V}$ be the category underlying the categorical Heisenberg algebra $H_{{\cat V}}$, and let $\bicat{F}_{{\cat V}}$ and $F_{{\cat V}}$ denote its categorical and classical Fock spaces, respectively. Let

$$\pi\colon H_{{\cat V}} \to K_0^{\mathrm{num}}(\bicat{H}_{{\cat V}},\, {\mathbb{k}})$$

be the canonical morphism.

Converse categorification conjecture. If $\bicat{F}_{{\cat V}}$ categorifies $F_{{\cat V}}$, then $\pi$ is an isomorphism.

The preceding theorem proves that an isomorphism of $\pi$ implies the categorification of the classical Fock space; this conjecture asserts the converse. Its status is not resolved in the supplied text.