Uniqueness conjecture for Laman graphs maximizing the number of realizations

Let $M_2(n)$ and $M_3(n)$ denote the maximum numbers of realizations in dimensions $2$ and $3$, respectively, among Laman graphs with $n$ vertices. For each $n\geqslant2$, let $G_{n,\max}$ be a Laman graph with $n$ vertices attaining the relevant maximum. Uniqueness conjecture. For each $n\geqslant2$ there is a unique Laman graph $G_{n,\max}$ with $n$ vertices and with the property

$$\mathrm{Lam}_2\left(G_{n,\max}\right)=M_2(n).$$

Similarly, there is a unique graph attaining $M_3(n)$. The conjecture concerns the classification of graphs with the maximal number of realizations; the paper reports uniqueness for $n\leqslant12$, while the assertion for all $n$ remains open.