Let W5 be the basic modal formula $\diamond \square \phi \rightarrow (\phi \rightarrow \square \phi)$ and let S4W5 be the smallest normal modal logic generated by the S4-axioms and W5. I want to find a frame class characterizing S4W5 but have failed so far.
I was able to show that the canonical frame $(W, R)$ of every normal logic containing W5 has the property $\forall x y (xRy \wedge yRx \rightarrow R(x) \subseteq R(y) )$, where $R(w) : = \{u \in W: wRu \}$. However this property seems too strong when it comes to proving that W5 is valid in all frames with that property.
W5 is of course valid in the class of Euclidean frames, so I guess euclidicity must play some role here. This is also indicated by the fact that the contrapositive of W5 is $\phi \wedge \diamond \neg \phi \rightarrow \square\diamond \neg \phi$, which seems to mix the formulas which are canonical for symmetry and Euclidicity.
Any help would be appreciated.