Condition keyconj under nonempty root intersection

Let $\Phi_A$ and $\Psi_{\lambda_{\mathbf c}}^+$ be the sets defined by

$$\Phi_A=\{\epsilon_i-\epsilon_j\mid i<j\}\cap\Phi^+\setminus\Phi_I,$$

and

$$\Psi_{\lambda_{\mathbf c}}^+=\{\beta\in\Phi^+\setminus\Phi_I\mid\langle\lambda_{\mathbf c}+\rho,\beta^\vee\rangle\in\mathbb Z_{>0}\}.$$

Suppose that $\Phi_A\cap\Psi_{\lambda_{\mathbf c}}^+\neq\emptyset$. The conjecture. Condition~ is satisfied, and consequently, Theorem~~(1)--(4) hold, if $r$ is odd or if $r$ is even and $\omega_i\neq0$ for some $0\leq i\leq k-1$. The condition is presented as a conjectural converse-type case to the preceding sufficient criterion, which proves Condition~ when the intersection is empty. The supplied text does not establish this assertion or give evidence resolving its status.