The zero-evaluation criterion for Bernstein's second theorem

Let $F$ be the system of sphere equations, and let $\widetilde{F}(\mathbf{t})$ be the system obtained by zero evaluations of the $\delta$-variables. Zero-evaluation criterion. The conditions of Bernstein's second theorem for the system $F$ are satisfied if and only if the system $\widetilde{F}(\mathbf{t})$ has solutions for every zero evaluation of the $\delta$-variables. The conjecture is motivated by experimental computations suggesting that checking zero solutions for only one choice of $\widetilde{F}(\mathbf{t})$, rather than all $d^{n-d-1}$ choices, suffices to verify Bernstein's conditions; no resolution is given in the source.