Let be $(M,g)$ a riemannian manifold with a singular riemannian $\mathcal{F}$ in $M$, see [1] the definition of singular riemannian foliation.

The riemannian metric on $M$ induces a distance on $M$, and consequentely a pseudo-distance on $M/\mathcal{F}$ by

$$d_{M/\mathcal{F}}(L_1,L_2):= \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\} $$ where the infimum is taken over all finite sequences $(p_1, p_2,\dots, p_n)$ and $(q_1, q_2, \dots, q_n)$ with $p_1=L_1, q_n\in L_2$ and $q_i,p_{i+1}$ belong to the same leaf.

Is the pseudo-distance $d_{M/\mathcal{F}}$ in fact a distance?