  We give an example of a reflected diffferential equation which may have
infinitely many solutions if the driving signal is rough enough (e.g. of
infinite $p$-variation, for some $p>2$). For this equation, we identify a sharp
condition on the modulus of continuity of the signal under which uniqueness
holds. L\'evy's modulus for Brownian motion turns out to be a boundary case. We
further show that in our example, non-uniqueness holds almost surely when the
driving signal is a fractional Brownian motion with Hurst index $H <
\frac{1}{2}$. The considered equation is driven by a two-dimensional signal
with one component of bounded variation, so that rough path theory is not
needed to make sense of the equation.
