Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that $d>deg(C)$. Denote by $C_r$ the reduced scheme associated to $C$. It is clear the normal sheaves, $N_{C|\mathbb{P}^3}$ and $N_{C_r|\mathbb{P}^3}$ are locally free. Also, recall under the closed immersion, $i:C_r \hookrightarrow C$ we have the canonical maps, $i^*:H^0(N_{C|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$ and $H^0(N_{C_r|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$.

The question is whether it is true that the image of the latter map is contained in the image of $i^*$.