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

Define

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

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 $t(n)$ ?