The maximal-rank conjecture for space curves in the range $g\leq Kd^{3/2}$

Let $d$ and $g$ be natural numbers. A smooth, connected curve in $\mathbb{P}^3$ has maximal rank if, for every integer $n\geq 0$, the restriction map

$$H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(n))\longrightarrow H^0(C,\mathcal{O}_C(n))$$

has maximal rank. The Hilbert scheme of $\mathbb{P}^3$ parametrizes subschemes with fixed Hilbert polynomial.

Maximal-rank conjecture. There exists a constant $K$ such that, for all natural numbers $d,g$ satisfying

$$g\leq Kd^{3/2},$$

there exists an irreducible component of the Hilbert scheme of $\mathbb{P}^3$ whose general element is a smooth, connected curve of degree $d$ and genus $g$ of maximal rank.

This conjecture was stated in 1985 and concerns the existence of components of the Hilbert scheme whose general space curve has the expected postulation. The paper's abstract says that the authors prove the statement, so its resolution should be checked against the paper's proof and publication status.