Let $0<q<1$ and $\varphi(q;x)=\displaystyle \prod_{j=0}^\infty (1+q^jx),\;x\geqslant 0.$ Consider the following functions: $$l_k(x;q):=\frac{q^{k(k-1)/2} x^k}{(1-q)(1-q^2)\dots (1-q^k)\varphi(x;q)}\quad k\in\mathbb{N}.$$

Is it true that, for all $2\leqslant m\in \mathbb{N}$ and all $x\geqslant 0,$ the following inequalities hold: $$l_{mk}(x;q)\leqslant l_k(x;q^m)?$$

The question arises in the study of the Lupas $q$-transform defined in https://doi.org/10.1016/j.camwa.2010.11.025. Inequalities of similar nature related to the limit $q$-Bernstein operators have been posted on the MO site previously in Inequality for functions on [0,1] and Inequality for functions on [0,1], continued, however, I have been unable to employ the similar approach to this one.