Let UNAMBIGUOUS-3DM be defined by analogy to UNAMBIGUOUS-SAT, i.e. as a promise problem version of three-dimensional matching where we may assume there is no more than one solution.
Is there a randomised reduction from SAT, as there is for UNAMBIGUOUS-SAT, so that a polynomial-time algorithm for UNAMBIGUOUS-3DM would imply NP=RP?
For example, it would suffice for there to be a parsimonious reduction from SAT to 3DM, but I haven't found one in the literature.
If so, does it still work when restricting to planar instances of 3DM, i.e. where the bipartite graph associated with the 3DM instance is planar?