Equivalence of US, SUS and SDP under the regularity assumption

Let $\xi$ be a point process satisfying Assumption (regularity-1), with the associated constant $c_{\xi}$. The properties (US), (SUS) and (SDP) are the conditions defined in the paper for this point process. Equivalence conjecture. Under Assumption (regularity-1), the assumptions in parts (c) and (d) of Theorem (main) are unnecessary: (US), (SUS) and (SDP) are equivalent. The conjecture would complete the converse implications in the theorem by removing the additional hypotheses currently required in parts (c) and (d).