Ralph Greenberg phublished in 1977 his thesis, written under the direction of Kenkichi Iwasawa, in an American Journal of Math paper entitled Iwasawa invariants of totally real fields. Back then, the so-called Main Conjecture was still open and Greenberg studied the vanishing of a certain invariant $\lambda_p$ partly for its own sake and partly because in certain cases it would imply the Main Conjecture. Little by little, people start to refer at Greenberg's condition $\lambda_p=0$ for totally real base fields as "Greenberg's conjecture" although he had not conjectured anything and simply studied the question.
The Main Conjecture was proven by Mazur--Wiles (1984), then reproven in greater generality by Wiles (1990), eventually proven a third time by Kolyvagin--Rubin (1992 or so) with much easier techniques. But Greenberg's question whether $\lambda_p=0$ is still alive and active, people are trying to prove it in general and the few results contained in the original thesis, albeit somehow interesting, got more or less forgotten.