The Lusternik-Schnirelmann category of the Lie groups $Sp(n)$.  Since $Sp(1) = S^3$, $\mathrm{cat}(Sp(1)) = 1$.  In the 1960s, P. Schweitzer proved that $\mathrm{cat}(Sp(2)) = 3$.  *Based on this*, a folklore conjecture emerged that in general $\mathrm{cat}(Sp(n)) = 2n-1$. In 2001, it was proved that $\mathrm{cat}(Sp(3)) = 5$, so maybe it's true?