Denjoy's multivariate Peano differential conjecture

Let $Q^d$ be the $d$-dimensional cube, and let $f$ be a function on $Q^d$. For an integer $k\geq 2$, write $D_x^k f$ for the Peano $k$-th differential at $x$ and $\partial_x^k f$ for the $k$-th differential in the classical sense. Denjoy's conjecture. If the Peano $k$-th differential $D_x^k f$ exists at every point $x\in Q^d$, then the $k$-th differential $\partial_x^k f$ in the classical sense exists almost everywhere. The one-dimensional converse is attributed to Denjoy; the multivariate assertion is stated as the corresponding consequence, and its resolution is recorded in the source as proved.