Quasifibration conjecture for the polynomial-representative projection

Let $A_d(m,n)$ be the polynomial-representative space and let $\Psi_d:A_d(m,n)\to\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ be the natural projection to the based algebraic-map space. Quasifibration conjecture. The map $\Psi_d$ is a quasifibration. The source presents this as a sufficient condition toward the homotopy-equivalence assertion for $\Psi_d$; although its fibres are described as contractible, the quasifibration property itself is not established.