Equality of sectional category and higher topological complexity for real projective spaces

Let $G_s=(\mathbb{Z}_2)^{\times(s-1)}$, let $\pi_s$ be the projective product covering considered above, and let $\phi_s\colon(S^m)^{\times s}\to J_{\operatorname{secat}(\pi_s)}(G_s)$ be a $G_s$-equivariant map satisfying

$$\phi_s(x_1,\ldots,x_{s-1},-x_s)=\sigma_1\cdots\sigma_{s-1}\cdot\phi_s(x_1,\ldots,x_{s-1},x_s).$$

Equality conjecture. An $s$-motion planning algorithm for $\operatorname{\mathbb{R}P}^m$ with $\operatorname{secat}(\pi_s)+1$ $s$-local rules can be constructed out of such a map. Consequently,

$$\operatorname{secat}(\pi_s)\geq\operatorname{TC}_s(\operatorname{\mathbb{R}P}^m),$$

and the previously established expression for $\operatorname{secat}(\pi_s)$ becomes an equality for every $s\geq2$. This identifies the sectional category of the projective product covering with higher topological complexity and would provide motion-planning algorithms with the asserted number of local rules.