Projection-map conjecture for polynomial representatives of algebraic maps

Let $A_d(m,n)$ and $\widetilde{A}_d(m,n)$ be the spaces of polynomial data representing based and free algebraic maps from $\mathbb{R}\operatorname{P}^m$ to $\mathbb{R}\operatorname{P}^n$, and let $\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ and $\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ be the corresponding algebraic-map spaces. Projection-map conjecture. The natural projection maps

$$\Psi_d:A_d(m,n)\to\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n),\qquad \Gamma_d:\widetilde{A}_d(m,n)\to\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$$

are homotopy equivalences. The conjecture would, together with Theorem II, imply the preceding higher-dimensional approximation conjecture. The authors note that the fibres of $\Psi_d$ are contractible, but that this alone does not establish a homotopy equivalence.