Conjectured unification of short-interval Goldbach and twin-prime results

Let $L$ denote the logarithm parameter used in the paper, and let $A$ and $B$ be fixed positive constants. For integers $r$ and $q$, let $\mathfrak{S}(r,q,a)$ denote the relevant Goldbach singular series, and let the starred maximum range over reduced residue classes $a\pmod q$ satisfying $(a-r,q)=1$. There exist absolute constants $0<\theta,\theta_{1},\theta_{2}<1$ such that, whenever

$$X_{1}\geq R\gg X_{1}^{\theta_{1}},\qquad X_{2}\geq Y\gg X_{2}^{\theta_{2}},\qquad R\ll Y^{\theta},$$

Conjectured unification. The estimate

$$\sum_{\substack{r\in [X_{1},X_1+R]\\2\mid r}} \sum_{q\leq Q} \: \sideset{}{^{\displaystyle *}}\max_{\substack{a \:(q) \\ (a-r,q)=1}} \Big| \sum_{\substack{p_2\in [X_{2},X_2+Y] \\ p_2\equiv a\;(q) \\ p_2+p_3=r }} \log p_2\log p_3 - \mathfrak{S}(r,q,a)Y \Big|\ll \frac{RY}{L^A}$$

holds for all $Q\ll YX_{2}^{-1/2}L^{-B}$. Here $r$ and $p_{2}$ range over short intervals of lengths $R$ and $Y$, respectively; the Goldbach equation $p_2+p_3=r$ may instead be replaced by the twin equation $p_2-p_3=r$. This proposed estimate would unify the paper's two classes of results, but the source gives no resolution and describes it as apparently beyond current methods.