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 $\Lambda(n)$ denote the von-monoglot function. Then
$\sum_{n=1}^{N} \Lambda(n)\Lambda(n+2) \sim c \frac{N}{\log^2 N} + O(1)$
where $c$ is the twin prime constant.
In other words the twin prime asymptotic holds and the error term is $O(1)$.