Summary: The low-energy description of wrapped M5 branes in compactifications of M-theory on a Calabi-Yau threefold times a circle is given by a conformal field theory studied by Maldacena, Strominger and Witten and known as the MSW CFT. Taking the threefold to be \({\mathbb{T}}_6 \) or \(\mathrm{K}3\times {\mathbb{T}}^2 \), we construct a map between a sub-sector of this CFT and a sub-sector of the D1-D5 CFT. We demonstrate this map by considering a set of D1-D5 CFT states that have smooth horizonless bulk duals, and explicitly constructing the supergravity solutions dual to the corresponding states of the MSW CFT. We thus obtain the largest known class of solutions dual to MSW CFT microstates, and demonstrate that five-dimensional ungauged supergravity admits much larger families of smooth horizonless solutions than previously known.