Collapsing limits conjecture for semi-flat $ALH_d^*$-gravitational instantons

Consider a sequence of pointed semi-flat $ALH_d^*$-gravitational instantons satisfying the assumption of Theorem, with parameters $c_i$ and $\tau_i$. Collapsing limits conjecture. The following pointed Gromov–Hausdorff limits occur:

$$c_i\rightarrow\infty \quad\Longrightarrow\quad \mathbb{R}^4$$

with the flat metric;

$$c_i\rightarrow C>0,\qquad \operatorname{Im}(\tau_i)\rightarrow\infty \quad\Longrightarrow\quad \mathbb{R}^2\times T^2$$

with the product of flat metrics, where the size of $T^2$ is strictly increasing with $C$; and

$$c_i\operatorname{Im}(\tau_i)\rightarrow\infty,\qquad c_i\rightarrow 0 \quad\Longrightarrow\quad \mathbb{R}^2$$

with the flat metric. These predictions describe the expected boundary behavior of the partially compactified pointed moduli space and the collapsing regimes suggested by the semi-flat metric.