Fulek's Question on the Extremal Function of L_3

Fulek defined a weight-five three-row $0$-$1$ matrix $L_3$ and asked whether $\mathrm{ex}(n, L_3) = O(n)$. It is: every $r \times s$ matrix avoiding $L_3$ has at most $27r + 2s$ ones, so $6n - 8 \le \mathrm{ex}(n,L_3) \le 29n$ for $n \ge 5$. The same argument covers an infinite family of light three-row patterns, verifying a conjecture of Pettie and Tardos on linear light patterns for that family.