Proof Writing & Problem Solving · Superchallenge problem

In a round-robin tournament with $7$ players, each pair plays once and there are no draws. Let $w_i$ and $l_i$ be the numbers of wins and losses of player $i$. A write-up proves $\sum w_i^2 = \sum l_i^2$ by double counting. What is the largest possible value of $\sum_{i=1}^{7} w_i^2$?