  We revisit, in an original and challenging perspective, the problem of
testing the null hypothesis that the mode of a directional signal is equal to a
given value. Motivated by a real data example where the signal is weak, we
consider this problem under asymptotic scenarios for which the signal strength
goes to zero at an arbitrary rate~$\eta_n$. Both under the null and the
alternative, we focus on rotationally symmetric distributions. We show that,
while they are asymptotically equivalent under fixed signal strength, the
classical Wald and Watson tests exhibit very different (null and non-null)
behaviours when the signal becomes arbitrarily weak. To fully characterize how
challenging the problem is as a function of~$\eta_n$, we adopt a Le Cam,
convergence-of-statistical-experiments, point of view and show that the
resulting limiting experiments crucially depend on~$\eta_n$. In the light of
these results, the Watson test is shown to be \emph{adaptively} rate-consistent
and essentially adaptively Le Cam optimal. Throughout, our theoretical findings
are illustrated via Monte-Carlo simulations. The practical relevance of our
results is also shown on the real data example that motivated the present work.
