Sufficiently large Euler-Characteristic provides zero first cohomology of coherent sheaf?

Question

$\newcommand{\F}{\mathcal{F}}$ Let $X$ be a projective curve over $k$, $k$ a field. Let $\F$ be a coherent sheaf on $X$. Is there a bound $b(X,\F) \in \mathbb{Z}$ depending on $X$ and $\F$ such that

$$\chi_k(\F) > b(X,\F) \quad \Rightarrow \quad \dim_k \operatorname{H}^1(X,\F) = 0?$$