Existence of primitive multiple lines with good cohomology

Let $d \geq 5$ be an integer congruent to $2$ modulo $3$. A primitive $d$-line is a primitive locally Cohen–Macaulay $d$-fold structure supported on a line $L\subset\mathbb{P}^3$; its type $e$ is defined by $\mathcal{I}_{L,C}/\mathcal{I}_{L,C}^2\cong\mathcal{O}_L(e)$. Write $\mathcal{I}_C$ and $\mathcal{I}_L$ for the ideal sheaves of $C$ and $L$, respectively, and let $L_d$ be the infinitesimal neighborhood defined by $\mathcal{I}_L^d$. Primitive multiple-line existence conjecture. There exist primitive $d$-lines satisfying all of the following: (i) a primitive $d$-line of type $e=\frac{d-2}{3}$ not contained in a surface of degree $d-1$; (ii) a primitive $d$-line of type $e=\frac{d+1}{3}$ with support $L$ such that

$$\mathrm{H}^0\bigl(\mathcal{I}_C(n)\bigr)=\mathrm{H}^0\bigl(\mathcal{I}_L^d(n)\bigr)\quad\text{for }n\leq d;$$

thus it is not contained in a surface of degree $d-1$, and every surface of degree $d$ containing it contains $L_d$; and (iii) a primitive $d$-line of type $e=\frac{d+4}{3}$ with support $L$ such that

$$\mathrm{H}^0\bigl(\mathcal{I}_C(n)\bigr)=\mathrm{H}^0\bigl(\mathcal{I}_L^d(n)\bigr)\quad\text{for }n\leq d+1.$$

These existence statements would establish sharpness of the proposed genus bound in the crucial case $d=s$, and provide the cohomological analogue of maximal rank for primitive multiple lines.