The rare radiative leptonic decay $B^+\to\ell^+\nu_\ell\gamma$, with $\ell=\mu,\,e$, is considered a golden channel to probe the internal structure of the $B^+$ meson within the Light-Cone Distribution Amplitude theoretical framework. In the Standard Model, it is predicted to have a Branching Ratio of 𝒪($10^{-6}$), but it has not yet been observed. The most stringent Upper Limit has been set by the Belle experiment, while such a search at a hadron collider was long deemed unfeasible. A novel strategy developed at the LHCb experiment exploits photon conversions $\gamma\to e^+e^-$ in the VErtex LOcator (VELO) material, enabling the reconstruction of the displaced $B^+$ vertex. Since the information on the final-state neutrino is lost, the analysis relies on the corrected mass of the $\ell^+e^+e^-$ system as the main observable. A critical aspect of the strategy is the modelling of backgrounds, dominated by the neutral meson decays $\pi^0\to\gamma\gamma$ and $\eta\to\gamma\gamma$. The LHCb dataset also provides sensitivity to the corresponding $B_c^+$ decay modes. The Run 2 analysis of the $B^+\to\mu^+\nu_\mu\gamma$ channel demonstrates competitive sensitivity and validates the feasibility of this new approach. A parallel analysis using Run 3 data is underway and, benefiting from the upgraded detector and improved trigger performance, targets both muon and electron channels, with the prospect of getting to the first observation of these $B^+_{(c)}$ decays.