The transposition problem is shown to be decidable for inverse monoid presentations of the form Inv <X; e=1>, where e is a Dyck word on the alphabet X∪X−1. If e is a, restricted Dyck word, the left conjugacy problem is solved as well, and the (left) conjugacy relation is shown to be the union of the transitive closure of the transposition relation with the universal relation on the semilattice of idempotents, a result which does not hold for a general Dyck word e.