We have studied the $P\to\gamma^\star\gamma^\star$ form factor in Resonance Chiral Theory, with $P = \pi^0\eta\eta'$, to compute the contribution of the pseudoscalar pole to the hadronic light-by-light piece of the anomalous magnetic moment of the muon. In this work we allow the leading $U(3)$ chiral symmetry breaking terms, obtaining the most general expression for the form factor up to $\mathcal{O}(m_P^2)$. The parameters of the Effective Field Theory are obtained by means of short distance constraints on the form factor and matching with the expected behavior from QCD. Those parameters that cannot be fixed in this way are fitted to experimental determinations of the form factor within the spacelike region. Chiral symmetry relations among the transition form factors for $\pi^0,\eta$ and $\eta'$ allow for a simultaneous fit to experimental data for the three mesons. This shows an inconsistency between the BaBar $\pi^0$ data and the rest of the experimental inputs. Thus, we find a total pseudoscalar pole contribution of $a_\mu^{P,HLbL}=(8.47\pm 0.16)\cdot 10^{-10}$ for our best fit (that neglecting the BaBar $\pi^0$ data). Also, a preliminary rough estimate of the impact of NLO in $1/N_C$ corrections and higher vector multiplets (asym) enlarges the uncertainty up to $a_\mu^{P,HLbL}=(8.47\pm 0.16_{\rm stat}\pm 0.09_{N_C}{}^{+0.5}_{-0.0_{\rm asym}})10^{-10}$.