Let $X$ and $Y$ be two Hausdorff uniform spaces. A surjective uniformly continuous map $f: X \to Y$ is a quotient map if for every map $g$ from $Y$ to a uniform space $Z$ such that $g \circ f$ is uniformly continuous, $g$ is uniformly continuous.
Let $\hat X$ and $\hat Y$ denote the completions of $X$ and $Y$ respectively, and let $\hat f : \hat X \to \hat Y$ be the unique uniformly continuous extension of $f$. Is $\hat f$ a quotient map?
If the answer is positive, a reference would be appreciated.