CAT(0) conjecture for the stratified Euclidean metric on polynomial space

Let ${{\normalfont\textsc{Poly}}}_d^{mt}$ be the space of all monic degree-$d$ complex polynomials up to translation, equipped with its stratified Euclidean metric. CAT(0) conjecture. The metric on ${{\normalfont\textsc{Poly}}}_d^{mt}$ is ${\normalfont\textsc{CAT}}(0)$. This would give nonpositive curvature for a natural piecewise Euclidean metric on polynomial space and, for each fixed $d$, would imply the corresponding local ${\normalfont\textsc{CAT}}(0)$ property of the dual braid complex and the ${\normalfont\textsc{CAT}}(0)$ property of the $d$-strand braid group. Its general status is not specified in the source.