The homotopy-equivalence conjecture for normalized algebraic-map spaces

Let $A_d^{\mathbb{C}}(m,n)$ be the space of normalized $(n+1)$-tuples of homogeneous complex polynomials of degree $d$ defining algebraic maps from $\mathbb{R}\operatorname{P}^{m}$ to $\mathbb{C}\operatorname{P}^{n}$ without a common real projective zero, and let

$$\Psi_d^{\mathbb{C}}:A_d^{\mathbb{C}}(m,n)\longrightarrow \operatorname{Alg}^{*}_{d}(\mathbb{R}\operatorname{P}^{m},\mathbb{C}\operatorname{P}^{n})$$

be the natural projection to the corresponding based algebraic map. Homotopy-equivalence conjecture. The map $\Psi_d^{\mathbb{C}}$ is a homotopy equivalence. The fibers are described in the paper as convex and contractible, which motivates the conjecture, but the paper does not establish that this projection is a homotopy equivalence.