Is it true that $x_{n+1}=x_{n-1}+2\log x_n=\operatorname{li}^{-1}(n)+O(\log n)$?

Consider the recurrence

$$ x_{n+1}=x_{n-1}+2\log x_n, $$

with positive initial values chosen so that the sequence remains positive and increasing.

Since this may be rewritten as

$$ \frac{x_{n+1}-x_{n-1}}{2}=\log x_n, $$

it resembles the centred-difference discretisation of the differential equation

$$ x'(t)=\log x(t). $$

The corresponding continuous solution satisfies

$$ \operatorname{li}(x(t))=t+C, $$

so one is naturally led to expect

$$ x_n\sim \operatorname{li}^{-1}(n). $$

My question is whether one can prove the stronger estimate

$$ x_n=\operatorname{li}^{-1}(n)+O(\log n). $$

I would also be interested in references on asymptotics of nonlinear second-order recurrences that can be viewed as centred-difference discretisations of first-order differential equations.