  Distortions of the Cosmic Microwave Background energy spectrum of the $\mu$
type are sensitive to the primordial power spectrum through the dissipation of
curvature perturbations on scales $k\simeq 50$ - $10^4\,\mathrm{Mpc}^{-1}$.
Their angular correlation with large-scale temperature anisotropies is then
sensitive to the squeezed limit of the primordial bispectrum. For inflationary
models obeying the single-field consistency relation, we show that the observed
$\mu T$ angular correlation that would correspond to the local shape vanishes
exactly. All leading non-primordial contributions, including all non-linear
production and projection effects, are of the "equilateral shape", namely
suppressed by $k^2/{\cal H}_f^2$, where ${\cal H}_f\simeq 10^{-1}\,{\rm
Mpc}^{-1}$ is the Hubble radius at the end of the $\mu$-era. Therefore, these
non-primordial contributions are orthogonal to a potential local primordial
signal (e.g. from multi-field inflation). Moreover, they are very small in
amplitude. Our results strengthen the position of $\mu$ distortions as the
ultimate probe of local primordial non-Gaussianity.
