Asymptotic conjecture for short weighted averages of prime-pair counts

Let $\pi_{2k}(x)$ count primes $p\leq x$ such that both $p$ and $p+2k$ are prime. Let $E=E(x)$ be a function satisfying

$$\log x=o(E).$$

Short-average prime-pair conjecture. As $x\to\infty$,

$$\frac{1}{\left\lfloor E\right\rfloor^2}\sum_{1\leq k\leq E}\left(\left\lfloor E\right\rfloor-k\right)\pi_{2k}(x)\sim\frac{x}{\log^2 x}.$$

The conjecture would provide the expected asymptotic for these weighted short averages, complementing the paper’s proved lower bounds. The source presents it as a likely conjecture and gives no resolution, so it remains open.