Let $M^n$ be a smooth compact Riemannian manifold with geodesically locally convex boundary and sectional curvature at least $-1$. Let $x\in M$ and $\varepsilon\in (0,1)$.
Does there exist a constant $C_n$ depending on the dimension $n$ only such that $$vol_{n-1}(B(x,\varepsilon)\cap \partial M)<C_n\varepsilon^{n-1}$$ where $B(x,\varepsilon)$ denotes the open ball of radius $\varepsilon$ centered at $x$?
Remark. A weaker statement would be sufficient for me: there exists a function $\tau_n(\varepsilon)$ which converges to 0 when $\varepsilon \to 0$ such that $$vol_{n-1}(B(x,\varepsilon)\cap \partial M)<\tau_n(\varepsilon).$$
