Infinitude conjecture for the second binomial gcd condition

For a positive integer $n$, define

$$G_2(n)=\gcd\left(\frac{1}{4n+1}{4n+1\choose n},(2n+1)(4n+1)\right).$$

Second binomial gcd infinitude conjecture. There are infinitely many positive integers $n$ such that

$$G_2(n)=1.$$

The paper presents this as the analogue of the first gcd infinitude conjecture and reports computational evidence, while no proof is supplied.