Look for a large gap in the distribution of primes. For this conjecture, the gap between $n!+2$ and $n!+n$ will suffice. Set $y = n!+2$ (which is composite) and set $m$ (which will be $\frac{x+y}{2}$, so $x$ will be $2m-y$ eventually) to be the largest composite so that there are no primes between $m$ and $y$. Then there will be primes between $x$ and $m$ and none between $m$ and $y$.
So $\pi(x) + \pi(y) = \pi(x) + \pi\left(\frac{x+y}{2}\right) > 2 \pi\left(\frac{x+y}{2}\right)$. So your conjecture will fail for infinitely many pairs $x$ and $y$.
Gerhard "This Belongs On Another Forum" Paseman, 2015.06.30
