Weak equivalence conjecture for polynomial and function spaces with prescribed patterns

Let $d$ be even, let $\Omega_{\langle d]}$ be the poset of allowed zero-divisor patterns, and let $\Theta\subset\Omega_{\langle d]}$ be a closed sub-poset. Let $\mathcal P_d^{\mathbf c\Theta}$ and $\mathcal F_{\leq d}^{\mathbf c\Theta}$ denote the polynomial and function spaces associated with the complementary pattern $\mathbf c\Theta$. For the embedding $\mathcal I$ defined by the cut-off formula in the source,

$$\mathcal I:\mathcal P_d^{\mathbf c\Theta}\to\mathcal F_{\leq d}^{\mathbf c\Theta},$$

Pattern-preserving weak equivalence conjecture. The embedding $\mathcal I$ is a weak homotopy equivalence.

The conjecture is presented as a basis for computing the cohomology of convex envelopes of traversing flows. The supplied text does not state a proof or a resolution.