Axiomatization of Modal Logic Squares with Distinguished Diagonal

Modal logics of squared Kripke frames with distinguished diagonal are considered. It is shown that many such logics, unlike ordinary two-dimensional products, cannot be axiomatized by formulas with finitely many variables. The method resembles that used to obtain a similar result for $\ge3$-dimensional products of modal logics. The proof uses, in particular, generalized Sahlquist formulas.