Weaker sufficient estimate for systems satisfying condition (d)

Let $\{f_i\}_{i=1}^n$ be a system of functions satisfying condition (d) with parameter $p\in[0,1/2)$. For each fixed $x\in X$, let $f_k^*(x)$ denote the decreasingly ordered values among $|f_j(x)|$. Weaker sufficient-estimate conjecture. There exist constants $\varepsilon_0\in(0,1/2)$ and $L>0$ such that

$$\int_X\sum_{1\le k\le n^{1/2+\varepsilon}}f_k^*(x)\,d\mu(x)\le \operatorname{const}\cdot n^{p+L\varepsilon}$$

whenever $\varepsilon\in(0,\varepsilon_0)$. The paper presents this estimate as a weaker statement whose proof would imply a weaker form of the preceding generalization conjecture; no proof or resolution is supplied here.