Embedding of the 1-skeleton of a Coxeter group into its Davis complex

Let $(W,S)$ be a Coxeter system and let $Sigma$ be the corresponding Davis complex. It is well-known that the Davis complex may be equipped with a piecewise Euclidean metric so that it is a proper, complete $text{CAT}(0)$ metric space. Being the 1-sceleton of the Davis complex, the Cayley graph of $W$ (with respect to the generating set $S$) canonically embeds into $Sigma$.

It might be obvious, but I was wondering if this embedding sends geodesic paths (with respect to the word metric) in the Cayley graph to geodesic paths in $Sigma$ (with respect to the piecewise Euclidean metric).