The divisibility and support descent conjecture

Assume that $F$ and $G$ satisfy conditions (1)--(5) of the remainder vanishing conjecture. Let $F^\circ$ be the remainder, and use the paper's definitions of the $w$-divisibility conditions and divisibility-and-support conditions (DSC). For the indicated integer ranges, suppose the relevant higher-index conditions hold.

The divisibility and support descent conjecture. The following implications hold: for $w=(0,1)$ and $i\in[m(n-m)(a-1)/a+1,m(n-m)]$, the $i$-th $w$-divisibility conditions imply the $(i-1)$-st; for $w=(1,1)$ and $i\in[m(n-m)(a-1)/a,m(n-m)]$, the same implication holds; and for $i\in[1,m(n-m)(a-1)/a]$, the $i$-th DSC imply the $(i-1)$-st divisibility conditions.

The source states that this conjecture implies the preceding remainder-vanishing conjecture, and hence the Jacobian conjecture. No resolution status is supplied.