An upper-bound conjecture for equivariant parametrized topological complexity of configuration spaces

Let $F(\mathbb{R}^2,n)$ denote the ordered configuration space of $n$ distinct points in the plane, and let $\mathrm{TC}^{\Sigma_s}[F(\mathbb{R}^2,s+1)\to F(\mathbb{R}^2,s)]$ denote the $\Sigma_s$-equivariant parametrized topological complexity of the forgetful fibration that omits one point. The upper-bound conjecture.

$$\mathrm{TC}^{\Sigma_s}[F(\mathbb{R}^2,s+1) \to F(\mathbb{R}^2,s)] \leq 3+s.$$

The preceding argument does not apply when $t=1$, because $B_{s+1}^{\Sigma_s}/Z$ is not torsion free and consequently every CW-complex of type $K(B_{s+1}^{\Sigma_s}/Z,1)$ is infinite dimensional. The stated upper bound is therefore left as a conjectural case requiring a different argument.