The weak Fitting-ideal generation conjecture

Retain the notation and assumptions of the preceding conjecture. Let $T$ be the distinguished element appearing in the source, let $M(d,\ell)$ be the specified sets of $(\tau,\nu)$-monomials, and let $\operatorname{Fitt}^{[1]}_{\mathcal{R}}(Z^0_{S'})$ denote the first shifted Fitting ideal.

Weak Fitting-ideal generation conjecture. The union of the sets

$$T^{1-r}M(t_2,r-1),\ T^{2-r}M(t_2-1,r-2),\ T^{3-r}M(t_2-2,r-3),\ \ldots,\ T^{t_2+1-r}M(0,r-1-t_2)$$

generates the ideal $\operatorname{Fitt}^{[1]}_{\mathcal{R}}(Z^0_{S'})$.

The source presents this as the weak form implied by the preceding, stronger conjecture. It also notes that the final exponent equals $(r-1)(r-2)/2$ and that the final set contains only the trivial monomial $1$. No resolution status is supplied.