Uniqueness of tangent cones under disk decomposition near line segments

Let $\Sigma$ be a minimal surface with area growth of $k$ planes. Suppose there is a uniform $\alpha<1$ such that, for every $R>R_0\gg1$, there exist line segments $L_i(R)$, with $1\leq i\leq m(R)<M$, and outside an $\alpha$-sublinearly growing neighborhood of $\bigcup_i L_i(R)$, the intersection $\Sigma\cap B_R$ is a union of disks.

Disk-decomposition uniqueness conjecture. Under these hypotheses, $\Sigma$ has a unique tangent cone at infinity.

This is proposed as a potential step toward resolving Meeks' uniqueness conjecture. It extends the setting in which all tangent cones are unions of planes with a common axis, while imposing disk structure away from a controlled neighborhood of finitely many line segments.