The coherent-family limit conjecture for the ultra-discretization of the affine geometric crystal

Let $\chi$ be the affine geometric crystal described above, with positive structure $\theta:{\mathcal V}_1\longrightarrow T$. Its ultra-discretization is the Kashiwara crystal ${\mathcal UD}(\chi,T,\theta)$.

Coherent-family limit conjecture. The crystal ${\mathcal UD}(\chi,T,\theta)$ is the limit of a coherent family of perfect crystals of type $\TY(G,1,2)$ in the cited work.

This conjecture identifies the ultra-discretization of the affine geometric crystal of type $D_4^{(3)}$ with a limit arising from coherent families of perfect crystals for the Langlands-dual algebra. The source does not provide evidence resolving the conjecture.