Stronger minimal-generator conjecture when the first format parameter is one

Assume that $r_1=1$. Let $W(\mu)$ be the representations associated with the weight parameters $\alpha$, $\beta$, $\gamma$, $a$, $b$, and $c$ for the generic ring $\hat R_{\mathrm{gen}}$.

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

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

Moreover, the first representation is redundant if $r_3=1$, and the second is redundant if $r_3>1$.

This is a stronger expectation than the six-generator conjecture. The source notes that the last representation is the variable $a_1$, so its contribution is completely understood, but gives no general proof of the asserted generation statement.