Disclaimer : the following reasoning is a physicist's one and as such may not be suitable for this website. Still it may give rise to potentially interesting insights so I ask it anyway.


I stumbled upon an article whose abstract claims that for any sufficiently large $ x $ and any $ \delta>0.525 $, one has $\pi(x+x^{\delta})-\pi(x)\gg\_{k}\dfrac{x^{\delta}}{\log^{k}x}$  

where $\pi(y)$ is the number of primes less than or equal to the real number $y$.


My idea is to make the rhs equal to 1, so that one would formally get : 


$ x^{\delta}\approx\log^{k}x $ 


Hence $ \delta\approx\dfrac{k\log\log x}{\log x} $ .


As the latter quantity attains a maximum of $\delta=1/e$ (as can be easily checked by calculus), this leads, assuming $ k $ can be taken equal to 1, to the following conjecture : 


For any large enough $ x $, one has $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ .


Note that this is stronger than what can be obtained assuming RH, but weaker than Cramer's conjecture.


My question is : is the considered conjecture supported by numerical computation ? Is there a better (hence "more rigorous") heuristics that leads to the same conclusion ? 



Is there some numerical evidence that $ \pi(x+x^{1/e})-\pi(x)\geq 1 $ for any large enough $ x $?
Turning my comment into an answer: your conjecture is supported by numerical computation, but much stronger ones are also supported: for instance, [Cramér's conjecture](https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture), based on models of the primes as a 'random set', suggests (with some slight modifications to curb its optimism) that the largest prime gaps are $O\left((\log n)^{2+\epsilon}\right)$: much smaller than $x^{1/e}$, and in fact smaller than $x^\epsilon$ for any $\epsilon$. [Terrence Tao has some discussion of these models](https://terrytao.wordpress.com/tag/cramers-random-model/) that you might be interested in.


On the flip side, even though your conjecture is probably much weaker than the actual asymptotics, it's beyond the current reach of provability; AFAIK the best known result without the RH is that there's a prime between $x$ and $x+x^{0.525}$, and improvements on that value come incredibly slowly; even getting it down to $x^{1/2}$ would be an incredible advance and (from what I understand of the way the bound has been improved over time) would take entirely new techniques.


