  It is now well understood that convex programming can be used to estimate the
frequency components of a spectrally sparse signal from $2m+1$ uniform temporal
measurements. It is conjectured that a phase transition on the success of the
total-variation regularization occurs when the distance between the spectral
components of the signal to estimate crosses $1/m$. We prove the necessity part
of this conjecture by demonstrating that this regularization can fail whenever
the spectral distance of the signal of interest is asymptotically equal to
$1/m$.
