Homotopy-equivalence conjecture for rational maps in the periodic critical-point locus

Let $d$ and $n$ be positive integers, let $\mathscr M_d$ be the space of degree-$d$ branched self-coverings of the sphere $S^2$, and let

$$\mathscr P_{d,n}=\{f\colon S^2\righttoleftarrow: f\text{ has two critical points of order }d,\text{ one of which has period exactly }n\}.$$

After giving $S^2$ its complex structure, write ${\mathbb P^1}$ and let ${\mathbf{Rat}}_d=\{f\in\C(z):\deg(f)=d\}$. Homotopy-equivalence conjecture. The inclusion

$$\mathscr P_{d,n}\cap{\mathbf{Rat}}_d\hookrightarrow\mathscr P_{d,n}$$

is a homotopy equivalence up to dimension $d$. The corresponding connectedness problem for the algebraic curve arising from this locus in the moduli quotient is open; this conjecture is motivated by the known homotopy equivalence up to dimension $d$ between ${\mathbf{Rat}}_d$ and $\mathscr M_d$, while the paper's main theorem establishes path connectedness of the larger topological space $\mathscr P_{d,n}$.