Say I have a Hadamard $d$-manifold $M$ with an upper Ricci curvature bound of $-b^2$. Write the volume form in polar exponential coordinates at $p\in M$ as $V(r,\theta) \, dr \, d\theta$, and similarly write the volume form of the simply connected $d$-manifold with constant curvature $-b^2$ as $V_{const}(r)\, dr\, d\theta$ in exponential coordinates.
Is it true that the ratio $V(r,\theta)/V_{const}(r)$ is bounded away from zero?
There's a true analogous statement for lower Ricci bounds, see e.g. Eq 4.2 of http://mail.math.ucsb.edu/~wei/paper/06survey.pdf.