2–Segal objects and the Waldhausen construction

In a previous paper, we showed that discrete version of the $S_\bullet$-construction gives an equivalence categories between unital 2-Segal sets and augmented stable double categories. Here, generalize this result to homotopical setting, by showing there is Quillen model category for objects Segal which given $S_\bullet$-construction. We show fits together with in case briefly discuss how it encompasses other known $S_\bullet$-constructions.