An "AGM-GAM" inequality

For positive real numbers $x_1,x_2,\ldots,x_n$ and any $1\leq r\leq n$ let $A_r$ and $G_r$ be , respectively, the arithmetic mean and geometric mean of $x_1,x_2,\ldots,x_r$.

Is it true that the arithmetic mean of $G_1,G_2,\ldots,G_n$ is never greater then the geometric mean of $A_1,A_2,\ldots,A_n$ ?

It is obvious for $n=2$, and i have a (rather cumbersome) proof for $n=3$.

It's a special case ($r=0$, $s=1$) of the mixed means inequality $$ M_n^s[M^r[\bar a]]\le M_n^r[M^s[\bar a]], \quad r,s\in \mathbb R,\ r<s, $$ where $M^s$ is the power mean with exponent $s$, see Survey on Classical Inequalities, p. 32, theorem 2.