Strongly solid $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic MASA

Using an extension of techniques of Ozawa and Popa, we give an example of a non-amenable strongly solid ${\mathrm{I}\mathrm{I}}_1$ factor $M$ containing an “exotic” maximal abelian subalgebra $A$: as an $A$,$A$-bimodule, $L^2(M)$ is neither coarse nor discrete. Thus we show that there exist ${\mathrm{I}\mathrm{I}}_1$ factors with such property but without Cartan subalgebras. It also follows from Voiculescu’s free entropy results that $M$ is not an interpolated free group factor, yet it is strongly solid and has both the Haagerup property and the complete metric approximation property.