Equivalent linear-algebraic formulation of the weighted initial-ideal conjecture

Let $P=\mathbb{C}[x,y,z]$ with weights $\operatorname{wt}(x)=1$, $\operatorname{wt}(y)=2$, and $\operatorname{wt}(z)=3$, and let $m\bsin\mathbb{N}_{>0}$. Set

$$R=\mathbb{C}[x,y,z]/(x,y)^{3m-2},\qquad M=(x,y,z)^{3m-2}/(x,y)^{3m-2}.$$

For a general $\operatorname{wt}$-homogeneous polynomial $f\bsin P$ of weight $3m$, multiplication by $f$ induces a vector-space map

$$f\cdot:R_w\longrightarrow M_{w+3m}.$$

Equivalent formulation. The map $f\cdot:R_w\to M_{w+3m}$ is an isomorphism for every $w\bsin\mathbb{N}$. This is presented as an equivalent linear-algebraic formulation of the weighted initial-ideal conjecture. It is intended to imply the primitive-multiple-line construction and hence the relevant maximum-genus cases; the source gives no resolution.