The complete homogeneous symmetric polynomials are defined as $$ h_k (x_1, \dots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq n} x_{i_1} x_{i_2} \cdots x_{i_k}. $$ For example, $$ h_3(x_1,x_2,x_3) = x_1^3 + x_2^3 + x_3^3 + x_1^2 x_2 + x_1^2 x_3 + x_2^2 x_3 + x_2^2 x_1 + x_3^2 x_1 + x_3^2 x_2 + x_1 x_2 x_3. $$ I'm interested in conditions as simple as possible (and ideally independent of $n$) that will suffice to prove that $$ h_1(x_1,\dots,x_n) > h_2(x_1,\dots,x_n) > h_3(x_1,\dots,x_n) > \cdots $$ for some given set of positive real numbers $\{ x_1, \dots, x_n \}$.

For example, do these inequalities hold if $h_1(x_1,\dots,x_n) \le 1$? Do they all hold if the first one $h_1(x_1,\dots,x_n) \ge h_2(x_1,\dots,x_n)$ holds?