  Let the crown $C_{13}$ be the linear $3$-graph on $9$ vertices
$\{a,b,c,d,e,f,g,h,i\}$ with edges $$E = \{\{a,b,c\}, \{a, d,e\}, \{b, f, g\},
\{c, h,i\}\}.$$ Proving a conjecture of Gy\'arf\'as et. al., we show that for
any crown-free linear $3$-graph $G$ on $n$ vertices, its number of edges
satisfy $$\lvert E(G) \rvert \leq \frac{3(n - s)}{2}$$ where $s$ is the number
of vertices in $G$ with degree at least $6$. This result, combined with
previous work, essentially completes the determination of linear Tur\'an number
for linear $3$-graphs with at most $4$ edges.
