DHT conjecture for K3 surfaces

Let $\mathcal{X}\to\Delta$ be an $L$-polarised Tyurin degeneration of K3 surfaces with general fibre $X$. Let $I$ denote the relevant lattice, let $e\in I$ be primitive and $m$-admissible in $L^{\perp}$, and define the mirror lattice

$$\check{L}=e^{\perp}_{L^{\perp}}/\mathbb{Z}e.$$

Let $Y$ be a $\check{L}$-quasi-polarised K3 surface. An allowable loop is a loop $\gamma\subset\mathbb{P}^1$ for which the quasi-polarisation is compatible with the elliptic fibration and the loop.

DHT conjecture. The surface $Y$ admits an elliptic fibration $\pi\colon Y\to\mathbb{P}^1$ and an allowable loop $\gamma\subset\mathbb{P}^1$ compatible with the $\check{L}$-quasi-polarisation, such that $\mathcal{X}\to\Delta$ and $\pi\colon Y\to\mathbb{P}^1$ form a mirror pair. Conversely, if an $L'$-quasi-polarised K3 surface $Y$ admits an elliptic fibration and an allowable compatible loop, if $\tau$ is $m$-admissible in $(L')^{\perp}$, and if

$$\check{L}'=\tau^{\perp}_{(L')^{\perp}}/\mathbb{Z}\tau,$$

then there exists an $\check{L}'$-polarised Tyurin degeneration $\mathcal{X}\to\Delta$ forming a mirror pair with $\pi\colon Y\to\mathbb{P}^1$.

This is a precise K3-surface version of the DHT philosophy relating Tyurin degenerations to codimension-one fibrations on Calabi–Yau manifolds. The source presents it as a conjecture; no resolution evidence is supplied here.