I was thinking about the Second Hardy-Littlewood conjecture for quite sometime (some of my posts are related to this). In one of my [earlier post](http://math.stackexchange.com/questions/959502/partial-proof-of-second-hardy-littlewood-conjecture-modified) I conjectured that the inequality ($\pi(x)$ denotes the prime counting function), $$\pi(x)+\pi(y)>2\pi\left(\dfrac{x+y}{2}\right)$$ is true. However, R. Israel showed that this inequality falls infinitely often. Basing upon the counter example that he gave I put forward the following two conjectures,

>**Conjecture 1** $\pi(x)+\pi(y)\le2\pi\left(\dfrac{x+y}{2}\right)$ for all sufficiently large $x$ and $y$.

>**Conjecture 2** $\pi(x)+\pi(y)<2\pi\left(\dfrac{x+y}{2}\right)$ for all sufficiently large $x$ and $y$.

It is trivial to see that the later conjecture implies the former. However, it seems that the former conjecture is true if the quantity $\dfrac{x}{y}$ be sufficiently large (assume $x\ge y$). 

So my question is that,

>Is there any proof of **Conjecture 1** or **Conjecture 2**?

Assuming there is no such proof at present can anyone suggest me some relevant literature which shed some light on proving this conjecture?