Extremal structure for saturation by cycles of length at least $\ell$

Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part, let $C_{\geq\ell}$ be the family of cycles of length at least $\ell$, and let $\Omega^{(\ell,k,n)}$ be the graph family defined in the paper. Long-cycle partite saturation conjecture. For $\ell>k\geq 3$, $\ell\geq 6$, and sufficiently large $n$,

$$sat(K_k^n,C_{\geq\ell})=kn-\ell+1+\left\lfloor\frac{\ell-2}{2}\right\rfloor^2+2\left(\ell-1-2\left\lfloor\frac{\ell-2}{2}\right\rfloor\right)\left\lfloor\frac{\ell-2}{2}\right\rfloor,$$

and all extremal graphs belong to $\Omega^{(\ell,k,n)}$. The conjecture proposes both an exact saturation number and a complete description of the extremal graphs; the supplied text provides no resolution evidence.