Is the "only if" direction of the following fact known?
For fixed sequences $(a)_i = a_1, \dots, a_r$, $(b)_i = b_1, \dots, b_r$ and $(c)_i = c_1, \dots, c_r$, the inequality $\prod_{i = 1}^r \left(\sum_j x^{a_i}_j y^{b_i}_j\right)^{c_i} \ge 1$ holds for all finite sequences of positive numbers $(x)_j, (y)_j$ if and only if it can be expressed as a finite product of positive powers of the Hölder inequalities $\left(\sum_j x^{a'}_j y^{b'}_j\right)^\lambda \left(\sum_j x^{a''}_j y^{b''}_j\right)^{1 - \lambda} \ge \sum_j x^{\lambda a' + (1 - \lambda) a''}_j y^{\lambda b' + (1 - \lambda) b''}_j$ and $\ell_p$-monotonicity inequalities of the form $\left(\sum_j x^a_j y^b_j\right)^\lambda \le \sum_j x^{\lambda a}_j y^{\lambda b}_j $ for $\lambda \in [0,1]$.
