Ganea conjecture

Ganea's conjecture is a claim in algebraic topology, now disproved. It states that
for all formula_2, where formula_3 is the Lusternik–Schnirelmann category of a topological space "X", and "S""n" is the "n"-dimensional sphere. 
The inequality 
holds for any pair of spaces, formula_5 and formula_6. Furthermore, formula_7, for any sphere formula_8, formula_2. Thus, the conjecture amounts to formula_10. 
The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, till finally Norio Iwase gave a counterexample in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with "X" a closed, smooth manifold. This counterexample also disproved a related conjecture, stating that 
for a closed manifold formula_12 and formula_13 a point in formula_12. 
This work raises the question: For which spaces "X" is the Ganea condition, formula_15, satisfied? It has been conjectured that these are precisely the spaces "X" for which formula_3 equals a related invariant, formula_17 