Model-independent spanning conjecture for q-bracket spaces

Let $\mathcal{Z}_q$ be the q-bracket space associated with the partition-function framework, let $\mathbb{P}$ be the space of partition observables, and let $\mathscr{H}$ and $\mathscr{J}$ be the distinguished subspaces used in the setup. Write $\langle\mathbb{P}\rangle_q$ for the q-brackets generated by $\mathbb{P}$, and similarly for intersections with these subspaces.

Model-independent q-bracket conjecture. One has

$$\mathcal{Z}_q=\langle\mathbb{P}\rangle_q=\langle\mathbb{P}\cap\mathscr{H}\rangle_q=\langle\mathbb{P}\cap\mathscr{J}\rangle_q.$$

The conjecture reformulates the preceding Bernoulli–Seki reduction independently of the choice of model. The supplied text gives no resolution status, so whether these equalities hold remains open here.