The induced $P_6$ planar Turán number conjecture

Let $P_6$ be the path on six vertices, and let ${\mathrm{ex}}_{\mathcal P}(n,P_6^{\mathrm{ind}})$ be the maximum number of edges in an $n$-vertex planar graph containing no induced copy of $P_6$. The induced $P_6$ planar Turán number conjecture. For every $n\geq 3$,

$${\mathrm{ex}}_{\mathcal P}(n,P_6^{\mathrm{ind}})=\begin{cases}3n-6,& n\leq 19,\\ \left\lfloor\frac{50}{17}(n-2)\right\rfloor+1,& n\geq 20. \end{cases}$$

This is presented as a conjecture for the more challenging problem of forbidding induced paths $P_k$ for $k\geq 6$; the claimed exact values remain open.