Equational compactness of bi-frames and projection algebras

We generalize D. Kelly's and K.A. Nauryzbaev's results of 1-variable and 2-variable equational compactness of complete distributive lattices satisfying the infinite distributive law and its dual ("bi-frames") to objects similar to monadic algebras (which we will call projection algebras). This will lead us to an example of bi-frame that is not 3-variable equationally compact, even for countable equation systems, thus solving a problem posed in 1978 by G. Grätzer. This example is realized as a certain complete sublattice of the complete Boolean algebra of regular open subsets of some Polish space.