The generalized outerplanar Turán conjecture for $P_6$

Let $f_{\mathcal{OP}}(n,P_6)$ denote the generalized outerplanar Turán number of the path $P_6$ on $6$ vertices, namely the maximum number of copies of $P_6$ in an outerplanar graph on $n$ vertices. Generalized outerplanar Turán conjecture for $P_6$.

$$f_{\mathcal{OP}}(n,P_6)=11n^2+\Theta(n).$$

This conjectures the best asymptotic value for the generalized outerplanar Turán number of a short path beyond the $P_5$ case established in the paper; determining the asserted linear-order error term remains open.