The sharp upper-bound conjecture for the Halin Turán number of the 6-cycle

Let $\operatorname{ex}_{\mathcal{H}}(n,C_6)$ denote the maximum number of edges in an $n$-vertex Halin graph containing no cycle of length $6$. The sharp upper-bound conjecture. For $n\geq 21$,

$$\operatorname{ex}_{\mathcal{H}}(n,C_6)\leq \frac{8}{5}(n-1).$$

This conjecture proposes the sharp upper bound for the Halin Turán number of the $6$-cycle; its status is not established by the supplied source context.