The symmetric-power criterion for surjectivity of the categorical Heisenberg map

Let ${\cat V}$ be the category underlying the categorical Heisenberg algebra, and let ${{\cat{S}}^{n}{\cat V}}$ be its $n$-th symmetric power. Write

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

for the canonical morphism, and let

$$\Sym^{n}K_0^{\mathrm{num}}({\cat V}) \to K_0^{\mathrm{num}}({{\cat{S}}^{n}{\cat V}})$$

be the canonical morphism.

Symmetric-power criterion conjecture. If the canonical morphism above is an isomorphism, then $\pi$ is an isomorphism.

The text explains that surjectivity of $\pi$ is related to additional homotopy idempotents arising in the perfect hull. It states that no general criterion is known and later says that a converse is proved, so the forward implication remains the conjectural direction.