The principal-polynomial support-reduction conjecture

Assume the hypotheses of the paper's supported-subset definition, including $(a,b,m,n)\in\mathcal{Q}$ and the associated objects $S,E^\circ,F^\circ,\alpha^\circ$. Fix $\ell\in[\mathrm{u}_F,\mathrm{v}_F]$, and let $P_1$ be the projection to the quotient by $L$. Suppose that $B$ is supported with respect to $(S,E^\circ,F^\circ,\alpha^\circ)$ and that $P_1(x_j)=0$ for $j\in[\ell+1,\mathrm{v}_F]$.

The principal-polynomial support-reduction conjecture. If $E^\circ$ is a principal polynomial, then either

$$P_1(x_\ell)=0$$

or there exists a proper subset of $B$ supported with respect to $(S,E^\circ,F^\circ,\alpha^\circ)$.

The source states that this conjecture implies the Jacobian conjecture. Its resolution status is not supplied.