The Maximal Rank Conjecture for general embedded curves

Fix integers $g,r,d$ with $\rho(g,r,d)\geq0$, and let $\mathfrak I_{d,g,r}$ be the Hilbert-scheme component whose general member is a smooth curve of degree $d$ and genus $g$ with general moduli. For an embedding $C\hookrightarrow\mathbb P^r$, write $\nu_n(C)$ for the restriction map

$$\nu_n(C):H^0(\mathbb P^r,\mathcal O_{\mathbb P^r}(n))\longrightarrow H^0(C,\mathcal O_C(n)).$$

Maximal Rank Conjecture. A general embedded smooth curve in $\mathfrak I_{d,g,r}$ has maximal rank: for every integer $n\geq1$, $\nu_n(C)$ is injective or surjective.

This predicts the dimensions of hypersurfaces containing a general embedded curve and unifies classical maximal-rank phenomena with canonical syzygy conjectures. The source does not give a complete resolution.