Distortions of the Cosmic Microwave Background energy spectrum of the µ type are sensitive to the primordial power spectrum through the dissipation of curvature perturbations on scales k 50 -10 4 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 µT 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 /H 2 f , where H f 10 −1 Mpc −1 is the Hubble radius at the end of the µ-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 µ distortions as the ultimate probe of local primordial non-Gaussianity. arXiv:1805.08775v2 [astro-ph.CO] 25 Aug 2018 1 We will consider only local fNL in this work. Conventionally, it is defined in terms of the Newtonian potential during matter domination, i.e. BΦ(k1, k2, k3) = −2fNLPΦ(k1)PΦ(k2) + 2 perms., where ζ = − 5Φ 3 [5,6]. Notice also that we use the notation of [7] for the comoving curvature perturbation.2 We use the loose language of "equilateral shape" and "local shape" to refer to the dependence of C µT that would be generated by equilateral or local primordial non-Gaussianity.