This Wikipedia article shows us what are the first twin primes. Here we state the notation for each twin prime pair writing $(q_n,2+q_n)$ for $n\geq 1$ (for example $q_1=3$ and $q_8=71$).
On the other hand this Wikipedia show what is the Bonse's inequality.
Question. I would like to know if, on assumption of some form of the Twin Prime conjecture, you can set a conjecture of the same kind of Bonse's inequality when one writes the elements of the sequence $(q_n)_{n\geq 1}$ instead of the full sequence of prime numbers in Bonse's inequality. See below my thoughts in next examples. Many thanks.
Example 1. One has that $$59^2<q_1q_2q_3q_4q_5q_6=3335145,$$ thus the similar inequality (than Bonse's inequality that one writes for twin primes) holds for $n=4$.
Example 2. Maybe is it possible to get a more sharper inequality, and with mathematical meaning, if one can state that there exists a $N$ and positive rational numbers $a,b$ such that $$(q_{n+1})^a<\left(\prod_{k=1}^n q_k\right)^b$$ holds for all $n>N$, on assumption that there are infinitely many twin primes.
That is: can you write, on assumption of a twin prime conjecture, a similar inequality of the same kind of Bonse's inequality now for twin primes?
In the Wikipedia's article is the reference to genuine Bonse's inequality.