  We show that the natural directed analogues of the KKL theorem [KKL88] and
the Eldan--Gross inequality [EG20] from the analysis of Boolean functions fail
to hold. This is in contrast to several other isoperimetric inequalities on the
Boolean hypercube (such as the Poincare inequality, Margulis's inequality
[Mar74] and Talagrand's inequality [Tal93]) for which directed strengthenings
have recently been established.
