  For the classical Shiryaev--Roberts martingale diffusion considered on the
interval $[0,A]$, where $A>0$ is a given absorbing boundary, it is shown that
the rate of convergence of the diffusion's quasi-stationary cumulative
distribution function (cdf), $Q_{A}(x)$, to its stationary cdf, $H(x)$, as
$A\to+\infty$, is no worse than $O(\log(A)/A)$, uniformly in $x\ge0$. The
result is established explicitly, by constructing new tight lower- and
upper-bounds for $Q_{A}(x)$ using certain latest monotonicity properties of the
modified Bessel $K$ function involved in the exact closed-form formula for
$Q_{A}(x)$ recently obtained by Polunchenko (2017).
