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Claim. Theorem 2 of Imre Z. Ruzsa, A just basis, Monatsh. Math. 109 (1990), no. 2, 145–151, gives a set $A$ of nonnegative integers such that every nonnegative integer is a sum $a+a'$ with $a,a'\in A$ and $\sum_{n\le N}\sigma(n)^2=O(N)$, where $\sigma(n)$ counts these representations. The function $f_2(n)$ of Problem 1192 counts ordered pairs, so $1\le f_2(n)\le2\sigma(n)$ whether $\sigma$ counts ordered or unordered pairs, and $\sum_{n\le x}f_2(n)^2\le4\sum_{n\le x}\sigma(n)^2\ll x$. If $A$ contains $0$, the set $A+1$ lies in the positive integers and has the same counts shifted by $2$, so it serves equally. Hence the answer is yes for $r=2$. The source card is ruzsa_1990_just_basis.

Covers. The case $r=2$. The cases $r\ge3$ are open.

Depends on. No page of this wiki.

Acceptance. Refereed: the publication in Monatshefte für Mathematik, whose issue the publisher's record dates June 1990; the page name uses the first day of that month. The site's remarks credit the result, but the site labels the problem OPEN, so the remarks are not acceptance.