This paper continues the program started by its author “to understand the combinatorial commutative algebra of the projective coordinate ring of the moduli stack \(\mathcal{M}_{C,\vec{p}}(\mathrm{SL}_2(\mathbb{C}))\) of quasi-parabolic \(\mathrm{SL}_2(\mathbb{C})\) principal bundles on a generic marked projective curve”. One finds “bounds for the degrees of polynomials needed to present these algebras by studying their toric degenerations”. A nice property which is proved (cf. Theorem 1.11) is “that the square of any effective line bundle on this moduli stack yields a Koszul projective coordinate ring”. For the proofs, among other ingredients, one constructs “a category of polytopes with term orders” and one shows “that many results on the projective coordinate rings of \(\mathcal{M}_{C,\vec{p}}(\mathrm{SL}_2(\mathbb{C}))\) follow from closure properties of this category with respect to fiber products.” The connections of this interesting paper with the previous literature are more complex than one can reasonably describe in a review without “forgetting” important aspects.