Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $C$ be a smooth projectively normal curve in $X$ and $\mathcal{L}$ be a line bundle on $X$ of the form $\mathcal{O}_X(D)$ for some (reduced) curve $D$. Let $k$ be a positive integer such that $\mathcal{L} \otimes_X \mathcal{O}_C$ is $k$regular (in the sense of CastelnuovoMumford regularity). Does there exist a positive integer $n$ such that the dimension of the kernel of the natural morphism from $H^0(\mathcal{L} \otimes_X \mathcal{O}_C (k)) \otimes H^0(\mathcal{O}_{\mathbb{P}^3}(n))$ to $H^0(\mathcal{L}\otimes_X \mathcal{O}_C(k+n))$ is strictly less than $h^0(\mathcal{O}_{\mathbb{P}^3}(n))$?
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