Maximal gcd conjecture for Lucas sequences

Let $(u_n)_{n\geqslant 0}$ be a Lucas sequence, and let $g_u(n)$ denote the gcd function associated with it. Assume that $\Delta_u\ne 1$. Then the maximal gcd conjecture asserts that

$$\max\,\{g_u(n)\,:\,n\leqslant x\}\sim x.$$

The conjecture is motivated by the expected abundance of positive integers $n$ dividing $u_n$, analogous to the distribution of Carmichael numbers; such integers would force the maximum to be asymptotic to $x$.