Semi-transition relations for semi-supermanifolds

Let $M$ be a semi-supermanifold represented by a semi-atlas of charts and semi-charts, with semi-transition functions $\Phi_{\alpha\beta}$ between local coordinate domains $U_\alpha$ and $U_\beta$. On pairwise overlaps, triple overlaps, and quadruple overlaps, these functions satisfy the corresponding idempotent composition relations.

Semi-supermanifold semi-transition conjecture. On $U_\alpha\cap U_\beta$,

$$\Phi_{\alpha\beta}\circ\Phi_{\beta\alpha}\circ\Phi_{\alpha\beta}=\Phi_{\alpha\beta}.$$

On $U_\alpha\cap U_\beta\cap U_\gamma$,

$$\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}\circ\Phi_{\gamma\alpha}\circ\Phi_{\alpha\beta}=\Phi_{\alpha\beta},$$ $$\Phi_{\beta\gamma}\circ\Phi_{\gamma\alpha}\circ\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}=\Phi_{\beta\gamma},$$ $$\Phi_{\gamma\alpha}\circ\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}\circ\Phi_{\gamma\alpha}=\Phi_{\gamma\alpha}.$$

On $U_\alpha\cap U_\beta\cap U_\gamma\cap U_\rho$, the analogous four cyclic relations hold:

$$\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}\circ\Phi_{\gamma\rho}\circ\Phi_{\rho\alpha}\circ\Phi_{\alpha\beta}=\Phi_{\alpha\beta},$$ $$\Phi_{\beta\gamma}\circ\Phi_{\gamma\rho}\circ\Phi_{\rho\alpha}\circ\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}=\Phi_{\beta\gamma},$$ $$\Phi_{\gamma\rho}\circ\Phi_{\rho\alpha}\circ\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}\circ\Phi_{\gamma\rho}=\Phi_{\gamma\rho},$$ $$\Phi_{\rho\alpha}\circ\Phi_{\alpha\beta}\circ\Phi_{\beta\gamma}\circ\Phi_{\gamma\rho}\circ\Phi_{\rho\alpha}=\Phi_{\rho\alpha}.$$

These relations are proposed as the noninvertible replacement for the usual transition-function cocycle conditions. The source gives no resolution or further justification, so the status is open.