Show that if $q$ is an odd prime there does not exist a positive integer $n>5$ such that $3^n$ divides $\binom{2^q}{2}-1$

Question

This problem is directly related to my most recent question. I would like to go a step further now and show the following is true

If $q$ is an odd prime then there does not exist a positive integer $n >5$ such that $$3^n\Bigg|\text{ } \binom{2^q}{2}-1$$

There is no deep motivation here just curiosity.