Celestial dual of conformal gravity MHV amplitudes: an OPE analysis

In an earlier paper [arXiv:2511.03669] we extracted the OPE of celestial CFT operator duals of positive helicity graviton and scalar particles from the Mellin transformed relevant MHV amplitudes of conformal gravity, realised as the bosonic subsector of the Berkovits-Witten theory. A soft theorem analysis of bulk MHV amplitudes established that this conformal gravity exhibits a chiral ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ symmetry on the celestial sphere with the associated $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ current algebra, which acquires a non-trivial central extension, unlike the Einstein gravity. Here we construct a $2d$ chiral CFT free-field realisation of the relevant chiral ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra in terms of three free scalars (${\phi }_i$) and three $({\beta }_i,{\gamma }_i)$ ghost pairs, and propose vertex operators for the positive-helicity graviton primary $G_{\mathrm{\Delta }}^{++}(z,\bar{z})$ as well as the scalar primary ${\mathrm{\Phi }}_{\mathrm{\Delta }}(z,\bar{z})$, and compute their OPEs. These OPEs reproduce exactly those obtained from the bulk conformal gravity MHV amplitudes, providing a concrete celestial dual description of its MHV sector.