Let $\mathbb{C}P^n$ denote the complex projective $n$-space, endowed with the metric that makes the quotient map $\pi : \mathbb{S}^{2n+1} \to \mathbb{C}P^n$ be a Riemannian submersion.
Given $p_1, p_2 \in \mathbb{C}P^n$, let $l = d(p_1, p_2)$. Let $g_i = \pi^{-1}(p_i)$ be the fibre over $p_i$ (a geodesic on the sphere). Is it true that $d(g_1, g_2) = l$?