Suppose we have a compact connected Lie group $G$ acting as isometries on a compact manifold $M^n.$ Then is it necessarily true that the Hausdorff dimension of the union of singular and exceptional orbits are no larger than the Hausdorff dimension of individual principal orbits? If not so, what about the union of singular orbits only?

It's known back in the 1950s through the works of Montgomery and Yang that in some cases, $\dim M^n\setminus M_{\text{principal}}\le n-2,$ the dimension of the complement of the union of principal orbits is at most $n-2.$ However, I'm not sure if they refer to Hausdorff dimension here. A search in the literature seems to indiate that there is no direct result on controlling $\dim M^n\setminus M_{\text{principal}}$ by the dimension of principal orbits.