Inductive refined Bachmann conjecture for the case of fewer zeros than depth

Fix $z,d,w\in\mathbb{Z}_{>0}$ with $z<d$, and assume that

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

for all $z'\leq z$, $d'<d$, and $w'<w$. For every $\mathbf{k}=(k_1,\ldots,k_d)\in\mathbb{Z}_{>0}^d$ and every word $\mathtt{W}=u_{k_1}u_0^{z_1}\cdots u_{k_d}u_0^{z_d}\in\mathcal{U}^{\ast,\circ}$ satisfying $\operatorname{zero}(\mathtt{W})=z$, $\operatorname{depth}(\mathtt{W})=d$, and $\operatorname{wt}(\mathtt{W})=w$, the inductive refined Bachmann conjecture asserts that

$$\zeta_q^{\mathrm{f}}(\mathtt{W})\in \operatorname{span}_{\mathbb{Q}}\left(S^{(2)}_{z,d,\mathbf{k}}\cup S^{(3)}_{z,d,\mathbf{k}}\right)+\operatorname{F}_{z,d,w}\subset\operatorname{F}_{z,d,w}.$$

In particular, $\operatorname{Fil}^{\mathrm{Z,D,W}}_{z,d,w}\mathcal{Z}_q^f\subset\operatorname{F}_{z,d,w}$. The paper says that its proofs provide evidence for this claim for $d\leq4$; the general inductive statement remains open.