We study the problem of optimal mixing of a passive scalar [Formula: see text] advected by an incompressible flow on the two-dimensional unit square. The scalar [Formula: see text] solves the continuity equation with a divergence-free velocity field [Formula: see text] with uniform-in-time bounds on the homogeneous Sobolev semi-norm [Formula: see text], where [Formula: see text] and [Formula: see text]. We measure the degree of mixedness of the tracer [Formula: see text] via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scales. For velocity fields with the above constraint, it is known that the decay of both mixing scales cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of [Formula: see text], which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato [Exponential self-similar mixing by incompressible flows, arXiv:1605.02090]. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.