# Can the approach followed in this article be used to improve the upper bounds for $H_{k},k>1$?

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 Elliott-Halberstam 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?
I've been to a series of talks by Maynard just recently (going beyond the "regular" presentation he gives). In the closing remarks, he spoke about Generalized EH conjecture and others, and he claimed (I hope I'm quoting him right) that the maximum that one can hope for in those sieve approaches is to get $p_{n+1)-p_{n}\leq 3$, but not $2$. This somehow contradicts the claimed results in the paper, although the results in the paper you cited seem to use different sieve. Anyways, I would be extremely careful with the paper you've linked until it will be refereed. – Asaf May 11 '14 at 19:47
I'm sorry but I don't get it. As the gap between two consecutive primes is necessarily even, if one gets $H_{1}\leq 3$, then $H_{1}\leq 2$. – Sylvain JULIEN May 11 '14 at 20:35
I think Asaf meant $H_1\leq 6$ instead of $H_1\leq 2$. This is now achieved by PolyMath8b under the generalized Elliott-Halberstam conjecture (finding two primes in the translates of any admissible triple). – GH from MO May 11 '14 at 21:28