  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 $s \leq 100$.
From the conjecture it would follow that $P(d,s)= P_{\text{MAX}}(d,s)$ for
$d=s$ and for every $d \geq 2s-1$.
