Gonçalves–Ramos equality conjecture for the linear-programming bound

Let $\boldsymbol v=(v_1,\ldots,v_{n-1})$ be a vector of increasing positive integers, and let $\operatorname{gap}(\boldsymbol v)$ and $\lambda_+(\boldsymbol v)$ be the quantities defined in the paper. The upper bound

$$\operatorname{gap}(\boldsymbol v)\leqslant \lambda_+(\boldsymbol v)$$

is attained in the cases listed in the paper: (i) all $v_i$ are odd; (ii) there are coprime integers $a,m\geqslant 1$ such that $a\{1,\ldots,m-1\}\subset\{v_1,\ldots,v_{n-1}\}$ and every integer in $\{v_1,\ldots,v_{n-1}\}\setminus a\{1,\ldots,m-1\}$ is not divisible by $m$; or (iii) $\boldsymbol v=a\boldsymbol v'$ for some integer $a\geqslant 1$, where $\boldsymbol v'$ satisfies (i) or (ii). Gonçalves–Ramos equality conjecture. Equality is attained in the upper bound if and only if one of these three conditions holds.

The conjecture records the cases suggested by the authors' linear-programming computations, which covered $n\leqslant 20$ and $\max(v_1,\ldots,v_n)\leqslant 40$. The supplied source gives no proof or resolution of the converse.