**EDIT.** Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $Ric\geq (n-1)\lambda$.
Fix a point $p\in M$. Let $B(r)$ denote the ball of radius $r$ centered at $p$.
Let $B_\lambda(r)$ denote a ball of radius $r$ in the $n$-dimensional space of constant sectional curvature $\lambda$.

**Question.** Is it true that the function $$r\mapsto \frac{vol(B(r))}{vol(B_\lambda(r))}$$ is decreasing?

A reference would be helpful.

Remarks. (1) For manifolds without boundary this is the Bishop-Gromov inequality.

(2) If one makes a stronger assumption that the sectional curvature of $M$ is at least $\lambda$ then, I think, the statement is also true. This is due to the fact that because of the convexity assumption $M$ is an Alexandrov space of curvature at least $\lambda$, and for them this theorem was proved by Burago, Gromov, and Perelman.