The average Hardy–Littlewood prime $k$-tuple conjecture

Let $\Lambda(n)$ be the von Mangoldt function, and for distinct integers $d_1,\ldots,d_k$ with $1\leq d_i\leq H$, let $\mathcal{D}=\{d_1,\ldots,d_k\}$. Define $E_k(x;\mathcal{D})$ by

$$\sum_{n\leq x}\prod_{i=1}^k\Lambda(n+d_i)=\mathfrak{S}(\mathcal{D})x+E_k(x;\mathcal{D}),$$

where $\mathfrak{S}(\mathcal{D})$ is the singular series. Average Hardy–Littlewood prime $k$-tuple conjecture. For $x\geq H$, one has

$$\mathop{\mathop{\sum_{d_1,..., d_k}}_{1 \leq d_i \leq H}}_{d_i\ \operatorname{distinct}} E_k(x,\mathcal{D})^2 \ll_k x^{1+\epsilon}H^k.$$

This average form was suggested as a possible weaker hypothesis sufficient for the short-interval distribution theorem, but the paper does not establish it.