Degeneracy formulas for upper degrees in DSC are conjectural

Prove that, in the DSC model, the following degeneracy formulas hold for all relevant generations: for (n−1)-simplices, D^{(n−1)}(0,n) = [(n − 1)(n + 2)]/2 and D^{(n−1)}(1,n) = n + 1 for all n ≥ 2; for (n−2)-simplices, D^{(n−2)}(0,n) = [(n − 2)(n − 1)n(3n + 11)]/24, D^{(n−2)}(1,n) = [n(n^2 + n − 4)]/2, D^{(n−2)}(2,n) = [(n − 1)n]/2, and D^{(n−2)}(3,n) = n for all n ≥ 3; and, for general d, the degeneracies of the largest and second-largest d-degrees satisfy D^{(d)}(M^{(d)}(n), n) = d + 2 for all n ≥ 1 and D^{(d)}(\widetilde{M}^{(d)}(n), n) = [(d + 1)(d + 2)]/2 for all n ≥ d + 2.

Background

The authors measured upper-degree degeneracies D^{(d)}(k,n) for small generations using computation and inferred closed-form expressions for several cases, including (n−1)- and (n−2)-simplices, and for the largest and second-largest d-degrees.

While these formulas match data for n ≤ 8, they remain unproven in general and are stated as conjectures to be established for all n.