Refined moment conjecture for greatest common divisors in Lucas sequences

Let $(u_n)_n$ be a non-degenerate Lucas sequence with constant coefficient $a_2=7$ and let $bb d$ be a positive integer. The notation is as in the conditional moment theorem for $(u_n)_n$.

Refined moment conjecture. As $x\to\infty$,

$$\sum_{n\leq x}\gcd(n,u_n)^\lambda=x^{\lambda+1}\exp\left(-\frac{\log x}{\log\log x}\left(\log\log\log x+\log\log\log\log x+o(1)\right)\right).$$

This conjecture refines the conditional order of magnitude for the moments of split greatest common divisors and is motivated by the analogy with refined asymptotics for pseudoprimes. The paper states it as a conjectural next step beyond its conditional and unconditional bounds.