Projection-map conjecture for spaces of real projective maps

Let $1\leq m<n$. The spaces $A_d(m,n)$ and $\tilde{A}_d(m,n)$ map to the spaces of based and unbased algebraic maps, respectively, via the projection maps

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

Projection-map conjecture. These projection maps are homotopy equivalences. This conjecture concerns the topology of spaces of equivariant algebraic maps between real projective spaces and predicts that the relevant projection maps capture their full homotopy type. The source attributes it to AKY1; its resolution is not specified here.