Refined Bachmann conjecture for formal qMZV filtrations

For positive integers $z,d,w$, define

$$\operatorname{F}_{z,d,w}:=\operatorname{Fil}^{\mathrm{Z,D,W}}_{z,d,w-1}\mathcal{Z}_q^f+\sum_{\substack{z'+d'=z+d-1\\0\leq z'\leq z}}\operatorname{Fil}^{\mathrm{Z,D,W}}_{z',d',w}\mathcal{Z}_q^f.$$

Refined Bachmann conjecture. For all $z,d,w\in\mathbb{Z}_{>0}$,

$$\operatorname{Fil}^{\mathrm{Z,D,W}}_{z,d,w}\mathcal{Z}_q^f\subset\operatorname{F}_{z,d,w}.$$

This strengthening is the paper's main approach to Bachmann's filtration conjecture. It is established in the ranges obtained by the paper's theorems, but no general resolution is supplied.