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 $y$.

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?

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