Surjectivity of modules and Castelnuovo-Mumford regularity

Question

Let $X$ be a smooth projective surface in $\mathbb{P}^t$ for some $t \ge 3$, $C$ a smooth curve in $X$ and $\mathcal{L}$ is a line bundle on $X$. Denote by $A_X$ (resp. $A_C$) the homogeneous coordinate rings of $X$ (resp. $C$). Denote by $i$ the natural closed immersion of $C$ into $X$. Suppose that $\Gamma_*(\mathcal{L})$ (resp. $\Gamma_*(i^*(\mathcal{L}))$ is generated (as an $A_X$ (resp. $A_C$)-module) in degree less than or equal to $n$. In other words, the natural morphisms from $H^0(\mathcal{L}(n)) \otimes H^0(\mathcal{O}_{\mathbb{P}^t}(m-n))$ (resp. $H^0(i^*\mathcal{L}(n)) \otimes H^0(\mathcal{O}_{\mathbb{P}^t}(m-n))$) to $H^0(\mathcal{L}(m))$ (resp. $H^0(i^*\mathcal{L}(m))$) for $m \ge n$ is surjective. Is it true that the natural morphism from $H^0(\mathcal{L}(k))$ to $H^0(i^*\mathcal{L}(k))$ is surjective for all $k \ge n$?