Sufficiency conjecture for multiplication profiles of hyperelliptic Jacobian point classes

Let $B$) be a hyperelliptic curve of genus $\ga$, with a marked Weierstrass point $q$, and let $\Theta_1^0(B)\subset \operatorname{Jac}(B)$ denote the relevant locus of point classes $[p-q]$. A multiplication profile is a sequence $(\ep_i)_{i=1}^{\infty}$ recording the sizes of the supports of the $q$-reductions of the multiples $i[p-q]$. Multiplication-profile sufficiency conjecture. Any sequence $(\ep_i)_{i=1}^{\infty}$ of non-negative integers satisfying the five listed conditions—namely $\ep_1=1$ and $0\leq \ep_j\leq \ga$; subadditivity $\ep_i+\ep_j\geq \ep_{i+j}$; the adjacent-difference and nonrepetition conditions; the prohibition on $\ep_{j+1}=\ep_j+1=\ep_{j+2}+1$ except when $\ep_j=\ep_2=0$; and the stated periodicity and symmetry after the first zero— is realized as the multiplication profile of some element $[p-q]\in\Theta_1^0(B)\subset \operatorname{Jac}(B)$ for some hyperelliptic curve $B$ of genus $\ga$. This would turn the necessary numerical conditions for multiplication profiles into a complete characterization, extending the paper's explicit classifications for cyclic covers and selected torsion orders. Whether every sequence satisfying these conditions is geometrically realizable remains open.