MacNeille completion and profinite completion can coincide on finitely generated modal algebras

Following Bezhanishvili and Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if ${\mathbb{A}}$ is a residually finite, finitely generated modal algebra such that HSP( ${\mathbb{A}}$ ) has equationally definable principal congruences, then the profinite completion of ${\mathbb{A}}$ is isomorphic to its MacNeille completion, and ? is smooth. Specific examples of such modal algebras are the free K4-algebra and the free PDL-algebra.