$\newcommand{\F}{\mathcal{F}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $f:X \to \operatorname{Spec}(k)$ be a projective scheme of dimension one over a field $k$. The Riemann-Roch equation for such curves is given, for instance in Liu's book (Thm. 7.3.26), and it states

$$ h^0(\F) - h^1(\F) = \deg_k \F + \chi(\ox)$$

for $\F$ an invertible $\ox$-module. Moreover, we have $h^1(\F) = h^0(\omega_f \otimes_{\ox} \F^{-1})$ where $\omega_f$ denotes the $1$-dualizing sheaf of $f$.

I am looking for a reference which states the following:

In the above situation, we have $$ h^0(\F) = \deg_k \F + \chi(\ox)$$ whenever $\deg_k \F$ is greater than a bound solely depending on $X$, for instance a constant multiple of $\chi(\ox)$.

Liu does give such a statement in his book, but he needs to assume that $X$ is a local complete intersection and, moreover, integral. But I wonder whether this is true in a more general setting.