General-position conjecture for measurable polynomial recurrence

Let $P=\{p_1,\ldots,p_k\}$ and $Q=\{q_1,\ldots,q_l\}$ be sets of polynomials. Let $\textup{\textsf{{WP}}}(P)$ and $\textup{\textsf{{WP}}}(Q)$ denote their associated Weyl-polynomial spaces. General-position recurrence conjecture. If

$$\textup{\textsf{{WP}}}(P)\setminus\textup{\textsf{{WP}}}(Q)\neq\emptyset,$$

then $Q$-measurable recurrence does not imply $P$-measurable recurrence. If in addition

$$\textup{\textsf{{WP}}}(Q)\setminus\textup{\textsf{{WP}}}(P)\neq\emptyset,$$

then $Q$-measurable recurrence and $P$-measurable recurrence are in general position. This is proposed as a converse-direction extension of the known implication from $\{n,2n\}$-recurrence to $\{n^2\}$-recurrence; the source also notes that the corresponding implication fails in the opposite direction in general systems. The conjecture is presented without a resolution.