Singular extensions of the trace and the relative Dixmier property in the type $\mathrm{II}_1$ factors

If $N \subset M$ is an inclusion of type $\mathrm{II}_1$ factors with $N'\cap M = \mathbf{C} I,$ we study the connection between the existence of singular states on $M$ which extend the trace on $N$ and the Dixmier approximation property in $M$ with unitaries in $N.$ We also prove the existence of singular conditional expectations from certain free product factors onto irreducible hyperfinite subfactors.