The Strong Conjecture for algebraic and noncrossing parking spaces

Let $W$ be a real reflection group of rank $n$ with Coxeter number $h$, let $V$ be its reflection representation, and let $k\geq 1$. Let $\theta_1,\dots,\theta_n$ be a homogeneous system of parameters of degree $kh+1$ carrying $V^*$, let $x_1,\dots,x_n$ be a basis of $V^*$, and write $\Theta=(\theta_1,\dots,\theta_n)$ and $\mathbf{x}=(x_1,\dots,x_n)$. Define $V^{\Theta}$ to be the variety cut out by $(\Theta-\mathbf{x})$. The groups $W$ and $\mathbb{Z}_{kh}$ act on $V^{\Theta}$, and ${{\sf Park}}^{\mathrm{alg}}_W(k)$ and ${{\sf Park}}^{NC}_W(k)$ denote the algebraic and noncrossing parking spaces. Strong Conjecture. For any such hsop and basis for which the $\mathbb{C}$-linear map induced by $x_i\mapsto\theta_i$ is $W$-equivariant, $V^{\Theta}$ consists of $(kh+1)^n$ distinct points of multiplicity one, with a $W\times\mathbb{Z}_{kh}$-equivariant identification $V^{\Theta}\cong {{\sf Park}}^{\mathrm{alg}}_W(k)$, and there is a $W\times\mathbb{Z}_{kh}$-equivariant bijection $V^{\Theta}\xrightarrow{\sim}{{\sf Park}}^{NC}_W(k)$. The rank-one case verifies the conjecture, and the paper gives further evidence; its validity for arbitrary reflection groups and hsops remains open.