  We show that the Liechti-Strenner's example for the closed nonorientable
surface in \cite{LiechtiStrenner18} minimizes the dilatation within the class
of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all
but the first coefficient of the characteristic polynomial of the action
induced on the first cohomology nonpositive. We also show that the
Liechti-Strenner's example of orientation-reversing homeomorphism for the
closed orientable surface in \cite{LiechtiStrenner18} minimizes the dilatation
within the class of pseudo-Anosov homeomorphisms with an orientable invariant
foliation and all but the first coefficient of the characteristic polynomial
$p(x)$ of the action induced on the first cohomology nonpositive or all but the
first coefficient of $p(x) (x \pm 1)^2$, $p(x) (x^2 \pm 1)$, or $p(x) (x^2 \pm
x + 1)$ nonpositive.
