The three-dimensional connected-generating-function odd qMZV conjecture

Let $X$ be a smooth threefold with equivariant Hilbert schemes and let $\operatorname{ch}_{k_i}$ be Chern-character components. Define the normalized connected generating functions $\left\langle\prod_i\operatorname{ch}_{k_i}\right\rangle^{\!\circ}$ and the algebra

$$\mathsf{oqZ}=\mathbb{Q}\left[\left(q\frac{d}{dq}\right)^lZ(2k+1)\right]_{k\geq1,\,l\geq0}\subset\mathsf{qMZV}.$$

Give the generators $\left(q\frac{d}{dq}\right)^lZ(2k+1)$ 3D weight $2k+1+l$ and depth $1$, and let $\mathsf{oqZ}_{\leq w,\geq d}$ denote the part of 3D weight at most $w$ and depth at least $d$.

Three-dimensional qMZV conjecture. Connected generating functions satisfy

$$\left\langle\prod_i\operatorname{ch}_{k_i}\right\rangle^{\!\circ}\in\sum_d\Bbbk_d\otimes_{\mathbb{Q}}\mathsf{oqZ}_{\leq 2+\sum_i(k_i+1),\geq d},$$

where the first subscript is an upper bound on 3D weight and the second specifies the depth bound.

This is a proposed structural description of connected three-dimensional Hilbert-scheme generating functions in terms of odd $q$-zeta values and their derivatives. The surrounding text also proposes, separately, that $\mathsf{oqZ}$ should be free on its generators; the candidate recorded here is the asserted containment for connected generating functions.