The historical highlights for this conjecture are: 

1923 : Hardy and Littlewood's classic paper, *Partitio Numerorum III*, elevates an obvious question to a conjecture.  More precisely, H & L note that $\pi(x+y) \leq \pi(x) + \pi(y)$ (*for large enough* $y$) is "forcibly suggested" by the data for $x,y \leq 200$; prove some upper and lower bounds on the (lim sup) densest packing of primes in an interval of length $x$; calculate the densest packing for $x=35, 59$ and $97$; and finish with the remark that "beyond $x=97$ it would seem that [the densest packing] falls further below $\pi(x)$, at least within any range in which calculation is practicable".  These speculative comments from the paper become known as a conjecture.

1973 : Ian Richards and his doctoral student Douglas Hensley explode the conjecture by showing that it contradicts the (much more plausible) prime k-tuplets conjecture 
 
2004 to today:   Green-Tao theorem and later developments lead to proofs of statements similar to the k-tuplets conjecture, giving hope that "moral certainty that H-L 1923 is false" can be replaced by "a proof that k-tuplets conjecture is true (and HL1923 is therefore false)". 

The $\pi(x+y)$ conjecture never had much evidence for it.  Hensley and Richards set out to disprove it by a computer calculation.  They ended up proving the existence of subsets $D$ of {1,2,...,$n$} denser than the first $n$ primes ($|D| > \pi(n)$) and with no congruence obstruction to $x + D$ being a set of primes for infinitely many $x$ (for no prime $p$ do the elements of $D$ fill all congruence classes modulo $p$).  Sets free of congruence obstructions are called "admissible prime constellations" and the other more credible conjecture of Hardy and Littlewood is that for any admissible constellation, infinitely many copies exist in the primes.  For $D$ this implies that for infinitely many $x$, 

$\pi(x+|D|) - \pi(x) \geq |D| > \pi (D)$.

The optimal example (according to http://www.opertech.com/primes/k-tuples.html ) is with $n = 3159, |D| = 447 > \pi(n) = 446$, so conjecturally,

$\pi(x+3159) > \pi(x) + \pi(3159)$  infinitely often, with the number of examples less than $N$ being of order $cN/(\log N)^{447}$  

Hensley and Richards' technique for constructing dense constellations of primes is to take all the primes and their negatives in a symmetrical interval $[-K,K]$, and delete enough of the first few primes to prevent congruence obstructions.  They proved that this construction is, infinitely often, denser than $\pi(2K+1)$.