The $T^{**}(r,n)$ extremal stability conjecture

Let $k,r,b$ be integers with $k\geq 2$, $r=2k+b$, and $3\leq b\leq 2k$. Let $T^{**}(r,n)$ be the three-block graph consisting of a complete bipartite block with parts of sizes $\left\lfloor\frac{n-r+2}{2}\right\rfloor$ and $\left\lceil\frac{n-r+2}{2}\right\rceil$, together with blocks $K_{2k}$ and $K_b$. $T^{**}(r,n)$ extremal stability conjecture. There exists an integer $n_0$ such that every $C_{2k+1}$-free graph $G$ on $n$ vertices with $n\geq n_0$ and

$$e(G)\geq\left\lfloor\frac{(n-r+2)^2}{4}\right\rfloor+\binom{2k}{2}+\binom{b}{2}$$

satisfies

$$\gamma_2(G)\leq 2\binom{k}{2}+\binom{\left\lfloor\frac b2\right\rfloor}{2}+\binom{\left\lceil\frac b2\right\rceil}{2},$$

and equality holds if and only if $G=T^{**}(r,n)$. This is the formal version of the preceding next-stability claim and is intended to identify the unique equality case for the stated edge threshold.