The equality of Lusternik–Schnirelmann category and cup-length for special orthogonal groups

For the special orthogonal group $\mathrm{SO}(n)$, let $\operatorname{cat}(\mathrm{SO}(n))$ denote its Lusternik–Schnirelmann category and let $\operatorname{cuplen}(\mathrm{SO}(n))$ denote its cup-length. The conjecture.

$$\operatorname{cat}(\mathrm{SO}(n))=\operatorname{cuplen}(\mathrm{SO}(n))$$

for all $n\geq 1$. This is presented as a long-standing conjecture; the surrounding discussion gives general inequalities but does not establish equality for all special orthogonal groups.