Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my question is the following :

Is the following version of Stokes theorem correct?

Let $M$ be a smooth submanifold without boundary of $\mathbb{R}^N$ of dimension $n\leq N$, and suppose that the topological boundary $\partial M$ of $M$ is the finite disjoint union of (smooth) submanifolds of $\mathbb{R}^N$, each of them of dimension $\leq n-1$. Then for all $n-1$ form on $\mathbb{R}^N$ with compact support, we have $\int_M d\omega = \sum_i \int_{B_i} \omega$, where $(B_i)$ are all the submanifolds of dimension $n-1$ in the decomposition of $\partial M$.

I am aware of a generalization of Stokes theorem that contains the cone, that is in the book of
*Partial Differential Equations 1* of Sauvigny, where the notion of "singular set with capacity zero" is described, but I admit I don't understand what it means. I suppose (but I am not sure) that the previous statement is true if one can show that submanifolds of dimension $\leq n-2$ in the topological boundary of $M$ are with capacity zero with respect to $M$ in the sense of Sauvigny. Is that correct?