Chiral $\mathrm{\Lambda }$-${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ symmetry of 3d conformal gravity

We propose mixed boundary conditions for 3d conformal gravity consistent with variational principle in its second-order formalism that admit the chiral $\mathrm{\Lambda }$-${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra as their asymptotic symmetry algebra. This algebra is one of the four chiral $\mathcal{W}$-algebra extensions of $\mathfrak{s}\mathfrak{o}(2,3)$ and is a generalisation of the chiral ${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_4$ algebra responsible for soft theorems of graviton MHV amplitudes in ${\mathbb{R}}^{1,3}$ gravity to the case of non-zero negative cosmological constant. The corresponding charges calculated using the modified covariant phase space formalism are shown to be finite and integrable, and realise this non-linear $\mathcal{W}$-algebra.