D-branes (or not) in the non-Abelian T-dual of the SU(2) WZW model

Following on from a previous article [1], we consider open strings in the non-Abelian T-dual of the $SU{\left(2\right)}_k$ WZW model, with respect to the vector $SU\left(2\right)$ isometry. Since in this case the dual theory has an exact CFT description, we look at the chiral algebra-preserving D-branes. The general conclusion is that a large set of branes, while consistent boundary conditions on the worldsheet, are sent to infinity in the target space. This suggests inconsistency of the strict non-Abelian duality with open strings, in this case.