  Celestial MHV amplitudes are comprised of non-distributional leaf amplitudes
associated to an AdS$_3$ leaf of a foliation of flat spacetime. It is shown
here that the leaf amplitudes are governed by the same infinite-dimensional
soft `$S$-algebra' as their celestial counterparts. Moreover, taking the soft
limit of the smooth three-point MHV leaf amplitude yields a nondegenerate
minus-minus two-point leaf amplitude. The two- and three-point MHV leaf
amplitudes are used to compute the plus-minus-minus leaf operator product
coefficients.
