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 (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((x+y)/2) > 2 pi((x+y)/2).  So your conjecture will fail for infinitely many pairs x and y.  

Gerhard "This Belongs On Another Forum" Paseman, 2015.06.30