A differentiable manifold with non-coinciding dimensions under the continuum hypothesis

Under the assumption of the continuum hypothesis we construct a differentiable $n$-manifold $M^{n,m}$, $4\leqslant n<m$, of dimension
$$ m-1\leqslant\dim M^{n,m}\leqslant m<m+n-3\leqslant\operatorname{Ind}M^{n,m}\leqslant m+n-1. $$
The space $M^{n,m}$ is perfectly normal and hereditarily separable.