It is known that if an irreducible curve $C$ is a local complete intersection in $\mathbb{P}^n$, then $\wedge^{n-1}\mathcal{N}\otimes\omega_{\mathbb{P}^n}$ is isomomorphic to the dualizing sheaf $\omega_C$, cf. Thm. 7.11 in Hartshorne's book.
Is it true that if $C$ is an irreducible Gorenstein curve in $\mathbb{P}^n$ (or even $C$ a canonical Gorenstein curve), then $$\deg(\mathcal{N})-deg(\mathrm{T}_{\mathbb{P}^n\vert_{C}})\leq 2g-2?$$ Here $g$ is the arithmetical genus of $C$ and I do not assume $C$ l.c.i.
