The maximum genus problem for locally Cohen-Macaulay space curves

Let $P_{\text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in ${\mathbb{P}}^3$ that is not contained in a surface of degree $<s$. A bound $P(d,s)$ for $P_{\text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $\mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $\mathcal{C}$ has good cohomological properties. With the aid of \emph{Macaulay2} we checked the validity of the conjecture for $s\le 100$. From the conjecture it would follow that $P(d,s)=P_{\text{MAX}}(d,s)$ for $d=s$ and for every $d\ge 2s-1$.