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 <a href="https://doi.org/10.1016/j.camwa.2010.11.025">this paper</a>. Inequalities of similar nature related to the limit $q$-Bernstein operators have been posted on the MO site previously in https://mathoverflow.net/questions/269740/inequality-for-functions-on-0-1
and https://mathoverflow.net/questions/286501/inequality-for-functions-on-0-1-continued, however, I have been unable to employ the similar approach to this one.