Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Let $Q$ be an affine monoid, $\Bbbk[Q]$ the associated monoid $\Bbbk$-algebra, and $\Bbbk[\overline{Q}]$ its normalization, where we let $\Bbbk$ be a field. In this paper, in the case where $\Bbbk[Q]$ is homogeneous (i.e., standard graded), a difference of the Hilbert series of $\Bbbk[Q]$ and $\Bbbk[\overline{Q}]$ is discussed. More precisely, we prove that if $\Bbbk[Q]$ satisfies Serre's condition $(S_2)$, then the degree of the $h$-polynomial of $\Bbbk[Q]$ is always greater than or equal to that of $\Bbbk[\overline{Q}]$. Moreover, we also show counterexamples of this statement if we drop the assumption $(S_2)$.