The microbundle Thom-space conjecture for constrained polynomial spaces

Let $\Theta$ be a closed constraint, and let $\mathcal B_d^\Theta$ denote the corresponding space of degree-$d$ polynomials satisfying the constraint. Let $\mathcal K_{d+2}(\Theta)$ be the subspace being collapsed in $\mathcal B_{d+2}^\Theta$. Microbundle Thom-space conjecture. For any closed $\Theta$, the quotient

$$\mathcal B_{d+2}^\Theta\big/\\,\mathcal K_{d+2}(\Theta)$$

is the Thom space of an orientable topological $2$-dimensional microbundle over $\mathcal B_d^\Theta$.

If true, this would provide a better geometric understanding of the spaces used in the preceding homological stabilization argument. The supplied text gives no resolution, so the conjecture remains open.