Lipschitz regularity at infinity and affine linearity conjecture

Let $X\subset\mathbb{C}^n$ be a pure $d$-dimensional entire complex analytic subset. Suppose that there exists a sequence $T=\{t_j\}_{j\in\mathbb{N}}$ of real positive numbers such that $t_j\to +\infty$ and $E=C_{\infty}^T(X)$ is a $d$-dimensional complex linear subspace of $\mathbb{C}^n$. Here, $C_{\infty}^T(X)$ denotes the tangent cone at infinity of $X$ with respect to $T$, and $X$ is LNE at infinity when its inner and outer distances are uniformly comparable outside a compact set. The affine-linearity conjecture. If $X$ is LNE at infinity, then $X$ is an affine linear subspace of $\mathbb{C}^n$. This extends the preceding result for Lipschitz regularity at infinity, which gives affine linearity under stronger assumptions; the conjecture concerns the corresponding conclusion under LNE at infinity.