Unique nearest-point conjecture for the stochastic phylogenetic variety

Let $T=12|34$, let $I$ and $\Omega$ be the parameter sets specified in the paper, and define

$$P_{0}:=\varphi _{T}(k_{0},1,k_{0},1,m_{0}).$$

For $(k_{0},m_{0})\in I\times\Omega$, consider the point

$$\varphi _{T}(\tilde{x}(k_{0},m_{0}),1,\tilde{x}(k_{0},m_{0}),1,1)\in\mathcal{V}_{T}^{+}.$$

Unique nearest-point conjecture. The distance from $P_{0}$ to $\mathcal{V}_{T}^{+}$ is

$$d(P_{0},\mathcal{V}_{T}^{+})=d\bigl(P_{0},\varphi _{T}(\tilde{x}(k_{0},m_{0}),1,\tilde{x}(k_{0},m_{0}),1,1)\bigr),$$

and this point is the unique point of $\mathcal{V}_{T}^{+}$ minimizing the distance to $P_{0}$. Although the local minimum has been established, the paper states that it cannot prove this global-minimum claim and presents it as suggested by evidence.