The symmetric-power conjecture for Hochschild homology of smooth proper dg categories

Let $\mathcal{T}$ be a smooth proper dg category, and let $\operatorname{Sym}^n\mathcal{T}$ denote its $n$th symmetric power in the sense of Ganter–Kapranov. Write $\operatorname{Sym}^{\bullet}$ for the symmetric algebra and let $t$ be a formal variable. The symmetric-power conjecture. One has

$$\bigoplus_{n\geq 0}\operatorname{HH}_{*}(\operatorname{Sym}^n\mathcal T)t^n \cong \operatorname{Sym}^{\bullet}\left(\bigoplus_{i\geq 1}\operatorname{HH}_{*}(\mathcal T)t^i\right).$$

This generalizes the Fock-space formula for Hochschild homology of symmetric powers and is motivated by the corresponding formula for Hilbert schemes of points on smooth projective surfaces. The conjecture is presented as a proposed generalization and no resolution is given here.