CAT(0) equivalence conjecture for polynomial space and the dual braid complex

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation with its stratified Euclidean metric, and let $K_d$ be the dual braid complex with its orthoscheme metric. CAT(0) equivalence conjecture. The stratified Euclidean metric on ${{\normalfont\textsc{Poly}}}_d^{mt}$ is ${\normalfont\textsc{CAT}}(0)$ if and only if the orthoscheme metric on $K_d$ is locally ${\normalfont\textsc{CAT}}(0)$. This conjecture would make the polynomial-space curvature conjecture equivalent to the local curvature conjecture for the dual braid complex; the source gives no resolution status.