Sun's base- expansion conjectures

Let $n,m,k$ be positive integers with $m\geq 2$. For a positive integer, its base-$m$ expansion is the representation using digits from $0$ through $m-1$; a non-zero digit is a digit in this expansion that is not $0$.

Sun's conjecture. (I) There are at least $k$ non-zero digits in the base-$m$ expansion of

$$\frac{m^k-1}{m-1}n.$$

(II) The sum of all digits in the base-$m$ expansion of

$$(m^k-1)n$$

\nis at least $k(m-1)$.

These assertions generalize Sun's results for binary expansions and for prime bases, giving digit-expansion lower bounds for products involving geometric sums and $m^k-1$.