Why is the Greedy Conjecture so difficult?
Let me first try to summarize what is known about the Greedy Conjecture.


Blum, Jiang, Li, Tromp, Yannakakis prove that the Greedy Algorithm gives a 4-approximation, and Kaplan and Shafrir show that it gives a 3.5-approximation for the Shortest Common Superstring problem.
A version of the greedy algorithm is known to give a 3-approximation (Blum, Jiang, Li, Tromp, Yannakakis). 
The Greedy Conjecture holds when all the input strings all of length at most $3$ (Tarhio, Ukkonen; Cazaux, Rivals) or $4$ (Kulikov, Savinov, Sluzhaev).
The Greedy Conjecture holds if the Greedy Algorithm happens to merge strings in some specific order (Weinard, Schnitger; Laube, Weinard).
The Greedy Algorithm gives a 2-approximation of the compression Tarhio, Ukkonen (which is defined as the total length of the input strings minus the length of the shortest common superstrting).
There is an extremely efficient implementation of the Greedy Algorithm Ukkonen.


I think that one of the reasons why it's hard to prove the Greedy Conjecture might be the following.
Most of the approaches to proving approximation guarantees of the Greedy Algorithm analyze the overlap graph (or, equivalently, the prefix graph) of the input set of strings. We know only some properties of these graphs (such as Monge and Triple inequalities), but these properties are provably not sufficient for proving the Greedy Conjecture (Weinard, Schnitger; Laube, Weinard).