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 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?