  We prove that any weakly non-collapsed RCD space is actually non-collapsed,
up to a renormalization of the measure. This confirms a conjecture raised by De
Philippis and the second named author in full generality.
  One of the auxiliary results of independent interest that we obtain is about
the link between the properties
  $\quad$- $\mathrm{tr}(\mathrm{Hess}f)=\Delta f$ on $U\subset\mathsf{X}$ for
every $f$ sufficiently regular, $\quad$- $\mathfrak{m}=c\mathscr{H}^n$ on
$U\subset\mathsf{X}$ for some $c>0$,
  where $U\subset \mathsf{X}$ is open and $\mathsf{X}$ is a - possibly
collapsed - RCD space of essential dimension $n$.
