Approximate posterior distributions for a range of L_max. The distributions have broad tails to high values because of the small number of large-scale modulation modes that are measured, and hence large cosmic variance. For L_max ≥ 3 the distributions are pulled away from zero by the significant octopole modulation signal observed, and are gradually move back towards zero as we include more modulation modes that are inconsistent with large τ_NL values. As shown in Fig. 20 the octopole has significant frequency dependence and is therefore unlikely to be physical.