Minimal-generator conjecture for the generic ring

Let $W(\mu)$ denote the isotypic representation introduced in the paper, and let $\alpha$, $\beta$, and $\gamma$ be partition parameters, with $a$, $b$, and $c$ the remaining weight parameters. The ring $\hat R_{\mathrm{gen}}$ is the generic ring associated to the format, and Proposition 10.1 identifies the corresponding six families of generating weights.

Minimal-generator conjecture. The ring $\hat R_{\mathrm{gen}}$ is generated by the six representations $W(\mu)$ corresponding to

$$\alpha=(1),\quad \beta=\gamma=a=b=c=0;$$ $$a=1,\quad \alpha=\beta=\gamma=b=c=0;$$ $$\beta=(1),\quad \alpha=\gamma=a=b=c=0;$$ $$b=1,\quad \alpha=\beta=\gamma=a=c=0;$$ $$\gamma=(1),\quad \alpha=\beta=a=b=c=0;$$ $$c=1,\quad \alpha=\beta=\gamma=a=b=0.$$

This would substantially reduce the generating set supplied by the known semigroup description. The source presents the assertion as an expectation, and no resolution is supplied here.