Bihomogeneity and Menger manifolds

It is shown that for every triple of integers (α,β,γ) such that α ⩾ 1, β ⩾ 1, and γ ⩾ 2, there is a homogeneous, non-bihomogeneous continuum whose every point has a neighborhood homeomorphic to the Cartesian product of Menger compacta μα × μβ × μγ. In particular, there is a homogeneous, non-bihomogeneous Peano continuum of covering dimension four.