Here is a stumper. Define $c_{00}$ as the space of all finitely-supported real sequences, i.e. all sequences $(a_n)_{n=1}^\infty\in\mathbb{R}^\mathbb{N}$ such that $a_n\neq 0$ for only finitely many $n\in\mathbb{N}$.
Question 1 (the simplified version). For which $q\in(1,\infty)$ does the inequality \begin{equation}\left(\sum_{n=1}^\infty|a_n|^q\right)^{1/q}\leq\sup_{n_1<n_2<\cdots\in\mathbb{N}}\sum_{k=1}^\infty\frac{|a_{n_k}|}{k}\end{equation} hold for all sequences $(a_n)_{n=1}^\infty\in c_{00}$?
Question 2 (the fully generalized version). Fix $1\leq p<\infty$. For which $q\in(p,\infty)$ does there exist a positive real number $C=C(p,q)$ such that \begin{equation}\tag{$*$}\left(\sum_{n=1}^\infty|a_n|^q\right)^{1/q}\leq C\sup_{n_1<n_2<\cdots\in\mathbb{N}}\left(\sum_{k=1}^\infty\left|\frac{a_{n_k}}{k}\right|^p\right)^{1/p}\end{equation} for all sequences $(a_n)_{n=1}^\infty\in c_{00}$?
Note that question 1 is just question 2 with $C=p=1$.
Also note: I would be very happy with partial answers. For instance, can we just find some $p$, $q$, and $C$ so that $(*)$ holds? Or so that $(*)$ fails to hold?