Hyperbolic 4-manifolds with perfect circle-valued Morse functions

We exhibit some (compact and cusped) finite-volume hyperbolic -manifolds with perfect circle-valued Morse functions, that is circle-valued Morse functions with only index 2 critical points. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one. An immediate consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers and rank.