Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.

Define

$$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$

Is it consistent with current data that for $n > 100 $ we have :

$$ t(n) <  \ln(n+2)^2 \sqrt {(n+2)} $$

??

Is there a stronger plausible boundary for $t(n)$ ?