  Generalising the chiral boundary conditions of $\mathbb{R}^{1,3}$ gravity for
AdS$_4$ gravity, we derive chiral locally AdS$_4$ solutions in the Newman-Unti
gauge consistent with a variational principle whose asymptotic symmetry algebra
we show, to be an infinite-dimensional chiral extension of
$\mathfrak{so}(2,3)$. This symmetry algebra coincides with the chiral
$\mathfrak{bms}_4$ algebra in the flat space limit. We posit this symmetry
algebra as the chiral version of recently discovered
$\Lambda$-$\mathfrak{bms}_4$ algebra. We postulate line integral charges from
the bulk AdS$_4$ gravity corresponding to this chiral symmetry algebra and show
that the charges obey the semi-classical limit of a $\mathcal{W}$-algebra that
includes a level $\kappa$ Kac-Moody $\mathfrak{sl}(2,\mathbb{R})$ current
algebra. Furthermore, using the standard tools of $2d$ CFT, we derive the
quantum version of this $\mathcal{W}$-algebra which may be denoted by
$\mathcal{W}(2;(3/2)^2,1^3)$.
