Mahler U-number product-dimension conjecture

For $m\geq 1$, let $U_m$ be the set of real $U_m$-numbers in Mahler's classification, and for $n\geq 1$ let $U_m^n$ denote their $n$-fold Cartesian product. The $U_m$ product-dimension conjecture.

$$\dim(U_m^n)=n-1,\qquad \dim_P(U_m^n)=n.$$

Liouville numbers are precisely the $U_1$-numbers, and the stated results for $U_1$ and for two factors motivate this simultaneous generalization. The conjecture remains open in the supplied text.