I've heard that Roger Heath-Brown has presented the following "conjecture" at several conferences, most likely to illustrate our poor understanding of the topic more than because he actually believes it to be true.
Let $\pi_2(x)$ denote the number of twin primes less than $x$. Then
$\pi_2(x) = c \int_{0}^{x}\frac{dt}{\log^2 t} + O(1)$
where $c$ is the twin prime constant.
In other words, the twin prime asymptotic holds with error term is $O(1)$.