Infinitude conjecture for the first binomial gcd condition

For a positive integer $n$, define

$$G_1(n)=\gcd\left(\frac{1}{2n+1}{2n+1\choose n},(2n+1)(2n+3)\right).$$

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

$$G_1(n)=1.$$

The source notes that this condition follows whenever $2n+1$ and $2n+3$ are both prime powers; consequently, the twin prime conjecture would imply this conjecture, but the source leaves the claim itself open.