Does uniform continuity of bounded continuous functions implies the same for all continuous functions on a uniform space?

Recently I came to know about Atsuji space from the paper. A metric space $X$ is called an Atsuji space if every real-valued continuous function on $X$ is uniformly continuous. Strikingly I have found in the above paper that, $X$ is an Atsuji space if and only if every bounded real-valued continuous function on $X$ is uniformly continuous.

I would like to ask whether the same can be concluded for a uniform space.

That is, can we conclude the following:

For an uniform space $(X,\mathcal U),$ every real-valued continuous function (w.r.t. the topology induced by $\mathcal U$) on $X$ is uniformly continuous if and only if every bounded real-valued continuous function on $X$ is uniformly continuous.

Unfortunately I failed to construct a counterexample and consequently some help.

• You should maybe ask for completeness of your uniform space. Metric Atsuji spaces are indeed complete, this comes from one of the equivalent definitions: every pseudo-Cauchy sequence with distinct terms in (X, d) has a cluster point (this might still be true for any uniform space). – M. Dus Aug 27 '18 at 9:30