  Let $R$ be a standard graded polynomial ring that is finitely generated over
a field of characteristic $0$, let $\mathfrak{m}$ be the homogeneous maximal
ideal of $R$, and let $I$ be a homogeneous prime ideal of $R$. Dao and
Monta\~{n}o defined an invariant that, in the case that
$\operatorname{Proj}(R/I)$ is lci and for cohomological index less than
$\dim(R/I)$, measures the asymptotic growth of lengths of local cohomology
modules of thickenings. They showed its existence and rationality for certain
classes of monomial ideals $I$.
  The following affirms that the invariant exists and is rational for rings $R
= \mathbb{C}[X]$ where $X$ is a $2 \times m$ matrix and $I$ is the ideal
generated by size two minors and is to our knowledge, the first non-monomial
calculation of this invariant.
