Cograph cotree lower-bound conjecture

Let $C$ be a cograph, and let $\widetilde{T}$ be the tree obtained from its cotree by erasing the leaves. Cograph cotree lower-bound conjecture.

$$\gamma_{\mathcal P}(C)\geq |E(\widetilde{T})|-\#\{\text{internal vertices of }\widetilde{T}\text{ labeled with the join or join operatation}\}.$$

The proposed bound would sharpen the loose lower bound obtained in the paper and would imply structural finiteness results for cographs of bounded algebraic co-rank. Its status is open in the source context.