If $q$ is a prime, $gcd(x(x+2),q\#)=1$ and $q < x < q^2$, doesn't it follow that $x,x+2$ are twin primes?

Question

I recently asked a question that was not well received. That's ok. I don't disagree with the ratings if my question is unclear. I want to verify the foundation of my reasoning.

Doesn't it follow if:

• $q$ is prime
• $q\#$ is the primorial for $q$
• $\gcd(x(x+2),q\#)=1$
• $q<x<q^2$

Then:

$x,x+2$ are twin primes.

Please let me know if I am making a mistake.