  The relationships between the invariants and the homological properties of
$I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ have been studied extensively over the
past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid
behaviours of their Betti numbers. In this work we establish a local cohomology
counterpart of their theorem. To this end, we make use of properties of
sequentially Cohen-Macaulay modules and we study a generalization of such
concept by introducing what we call partially sequentially Cohen-Macaulay
modules, which might be of interest by themselves.
