HypothesiX residue-pairing bound conjecture

For a squarefree integer $Q$ with $6\mid Q$, let $U_Q$ be the set of residues $r\bmod Q$ such that both $r$ and $r+2$ are coprime to $Q$. Define the residue-pairing bound

$$B_Q(x)=\sum_{r\in U_Q}\min\left(\pi(x;Q,r),\pi(x;Q,r+2)\right),$$

where

\pi(x;Q,a)=\\#\\{p\leq x:p\equiv a\pmod Q\\}.

Also let $\pi_2(x)$ denote the twin-prime counting function. HypothesiX's residue-pairing conjecture. For all $x\geq 7$ and every squarefree $Q$ with $6\mid Q$,

$$\pi_2(x)\leq B_Q(x)+2.$$

The conjecture is machine-generated and is described as nearly as difficult as the parity problem; only a weaker form is proved in the paper.