Equivalence of validities for symmetric-transitive versus up/down-symmetric frame classes

Determine whether the class of symmetric-and-transitive frames Csym∩Ctra validates exactly the same set of formulas as the class Cusym∧Cdsym of frames that are both up-symmetric and down-symmetric; equivalently, decide whether Log(Csym∩Ctra) equals Log(Cusym∧Cdsym).

Background

The authors show several equalities between logics determined by standard and up/down versions of reflexivity, symmetry, and transitivity. However, the relationship between symmetric-transitive frames and those that are simultaneously up-symmetric and down-symmetric remains unsettled.