Higher-dimensional Mostovoy conjecture for algebraic maps between real projective spaces

Let $m,n,d$ be integers with $2\leq m<n$ and $d\geq1$. Write $\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ and $\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ for the free and basepoint-preserving degree-$d$ algebraic-map spaces, and let $\operatorname{Map}_{[d]_2}$ and $\operatorname{Map}_{[d]_2}^*$ denote the corresponding components of free and based continuous maps. Define $D(d;m,n)=(n-m)(\lfloor (d+1)/2\rfloor+1)-1$. Higher-dimensional Mostovoy conjecture. The inclusion maps

$$j_{d,\mathbb{R}}:\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)\to\operatorname{Map}_{[d]_2}(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$$

and

$$i_{d,\mathbb{R}}:\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)\to\operatorname{Map}_{[d]_2}^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$$

are homotopy equivalences through dimension $D(d;m,n)$ when $m+2\leq n$, and homology equivalences through the same dimension when $n=m+1$. This extends Mostovoy's one-dimensional-domain theorem; the paper states the conjecture but does not prove it, while Theorem II supplies related finite-dimensional approximation results.