Large Complex Structure Limit of Calabi-Yau family and uniqueness of limit

Question

Let $\mathcal X$ be a smooth complex manifold of dimension $n+1$. We say $\mathcal X \to ∆$ is a large complex structure limit if and only if it’s maximal unipotent degeneration .

$T: H^n(\mathcal X_s, \mathbb Z) → H^n(\mathcal X_s, \mathbb Z)$ is maximal quasi-unipotent, i.e.,

$∃s$ s.t. $(T^s − id)^{n+1} = 0$, $(T^s − id)^n\neq 0$

Here $T$ is the monodromy operator. Take $\mathcal X_s\cong X$ are Calabi-Yau varieties. We call $\mathcal X\to \Delta$ is the CY degeneration of $X$. This definition is due to the work of Griffiths–Landman–Grothendieck-Katz and P.Deligne

Let $X ⊂ \mathbb P^n$ be a CY variety and there exists large complex structure degeneration of $X$ then when the limit $\mathcal X_0$ is unique?

For example the uniqueness of the $\mathcal X_0$ may not be true if $X$ is complete intersection in toric orbifold.

I have a conjecture that the uniqueness of the solution of canonical metric(relative Kahler-Einstein metric ) on degeneration of CY family correspond to uniqueness of $\mathcal X_0$. In fact this is the motivation of my question

Answer 1

In Physics, the large complex structure limit of $X$ is dual to a large volume limit of the mirror Calabi-Yau $X^v$. More precisely, different phases of $X^v$ connected by flops correspond to different large complex structure limits on $X$. So I would conjecture that this is true if and only if the extended Kaehler cone on $X^v$ is just the ordinary Kaehler cone (as is true for all (I think) cases with $h^{1,1}(X^v) = 1$ but typically fails, e.g. for most toric hypersurfaces).