Universal $n$-soft maps of $n$-dimensional spaces in absolute Borel and projective classes

Let $\mathscr C$ be one of the absolute Borel classes ${\mathscr M}_\alpha$, ${\mathscr A}_\alpha$, $1\le\alpha<\omega_1$ or one of the absolute projective classes ${\mathscr P}_k$, $k\ge1$. A map of an $n$-dimensional space $X\in\mathscr C$ onto the Hilbert cube which is an $n$-soft map in Shchepin's sense and universal in the class of maps of spaces of dimension smaller that or equal to $n$ from the class $\mathscr C$ into separable metrizable spaces is constructed.