  In this note we examine the following random graph model: for an arbitrary
graph $H$, with quadratic many edges, construct a graph $G$ by randomly adding
$m$ edges to $H$ and randomly coloring the edges of $G$ with $r$ colors. We
show that for $m$ a large enough constant and $r \geq 5$, every pair of
vertices in $G$ are joined by a rainbow path, i.e., $G$ is {\it rainbow
connected}, with high probability. This confirms a conjecture of Anastos and
Frieze [{\it J. Graph Theory} {\bf 92} (2019)] who proved the statement for $r
\geq 7$ and resolved the case when $r \leq 4$ and $m$ is a function of $n$.
