Extremal construction conjecture for $\alpha_{1,k}$

For an integer $k\geq 3$, let $\alpha_{1,k}$ be the asymptotic extremal constant studied in the paper, and let $J_r$ and $K_r$ denote the indicated graph constructions. Extremal construction conjecture for $\alpha_{1,k}$. One has

$$\alpha_{1,k}=\frac{4k}{4k-1}.$$

Moreover, the maximum is achieved by a blowup of

$$G=\overline{kJ_2\cup (2k-1)K_1}=J_{2k-1}\vee \overline{kJ_2}.$$

This gives the proposed exact value and extremal construction corresponding to the conjectured tightness of the lower bound for $k\geq 3$. The supplied context does not state whether the conjecture has been resolved.