Existence and uniqueness of associative submanifolds with three K3 ends

Let $X$ be a K3 manifold, let $\alpha_1,\alpha_2,\alpha_3$ be $-2$ classes on $X$ satisfying

$$\alpha_1+\alpha_2+\alpha_3=0,$$

and let $H\subset H^2(X)$ be a maximal positive subspace corresponding to a hyperkähler structure. Write $v_i$ for the projection of $\alpha_i$ to $H$, and assume that each pair $(\alpha_i,H)$ is irreducible. Existence and uniqueness conjecture. There is an associative submanifold

$$\Pi\subset X\times \mathbb{R}^3$$

with three ends asymptotic to $\Sigma_i\times\mathbb{R}^+v_i$, where $\Sigma_i$ is the complex curve representing $\alpha_i$ for the complex structure defined by $v_i$; moreover, $\Pi$ is unique up to translations of $\mathbb{R}^3$. This conjecture predicts a canonical associative submanifold joining the three complex curves determined by the projected $-2$ classes, with uniqueness modulo the natural translation symmetry.