Let $(a,b)$ be a pair of coprime positive integers with $a$ being even. Are these conditions sufficient to prove that there exist infinitely many positive integers $n,$ such that $(a^n+1,b^n+1)=1$ ?
I think that this sort of question was originally asked by Ailon and Rudnick, but they use $-1$ instead of $+1$ and asked if $\gcd(2^n-1,3^n-1)=1$ for infinitely many $n$. In this setting, for more general $a$ and $b$, the right question/conjecture would be $\gcd(a^n-1,b^n-1)=\gcd(a-1,b-1)$. They prove something stronger if you replace $\mathbb Z$ with $\mathbb C[t]$. Here's the Ailon-Rudnick paper:
N. Ailon and Z. Rudnick, ‘Torsion points on curves and common divisors of $a^k-1$ and $b^k-1$, Acta Arith., 113 (2004), no. 1, 31–38 (MSN).
There have been lots of articles on this and related problems. For example, a Google search on "Ailon Rudnick gcd" brings up recent articles such as:
On some extensions of the Ailon–Rudnick theorem, A Ostafe - Monatshefte für Mathematik, 2016 (MSN)
On a variant of the Ailon–Rudnick theorem in finite characteristic. D Ghioca, LC Hsia, TJ Tucker - New York Journal of Mathematics, 2017 (MSN)
Greatest common divisors of iterates of polynomials, LC Hsia, T Tucker - Algebra & Number Theory, 2017 (MSN)