In http://arxiv.org/pdf/1405.0682.pdf, the author gives a conditional proof of the twin prime conjecture under both a generalized version of the ElliottHalberstam conjecture and a hypothesis on the main terms of the GPY sieve. My question is: can a similar approach provide better upper bounds for the quantities $H_{k},k>1$ where $H_{k}:=\lim\inf_{n\to\infty} p_{n+k}p_{n}$ than those currently obtained thanks to the Polymath8b project?
Thanks in advance.
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