For any two points $p$, $q$ in $Y$, it is a general fact that a minimizing geodesic in $C(Y)$ connecting $p$ and $q$ must be of the form $(r(t),\gamma(t))$ where $\gamma(t)$ is a geodesic in $Y$, and $r$ is a universal function of $d_{Y}(p,q)$ and $t$ determined by elementary trigonometry. By recent result of Colding-Naber [7] we know $C(Y^{reg})$ is geodesically convex in $C(Y)$, so the lemma follows. ∎