  We study the Andrews-Schilling-Warnaar sum-sides for the principal characters
of standard (i.e., integrable, highest weight) modules of $\mathrm{A}_2^{(1)}$.
These characters have been studied recently by various subsets of Corteel,
Dousse, Foda, Uncu, Warnaar and Welsh. We prove complete sets of identities for
moduli $5$ through $8$ and $10$, in Andrews-Schilling-Warnaar form. The cases
of moduli $6$ and $10$ are new. Our methods depend on the Corteel-Welsh
recursions governing the cylindric partitions and on certain relations
satisfied by the Andrews-Schilling-Warnaar sum-sides. We speculate on the role
of the latter in the proofs of higher modulus identities. Further, we provide a
complete set of conjectures for modulus $9$. In fact, we show that at any given
modulus, a complete set of conjectures may be deduced using a subset of "seed"
conjectures. These seed conjectures are obtained by appropriately truncating
conjectures for the "infinite" level. Additionally, for moduli $3k$, we use an
identity of Weierstrass to deduce new sum-product identities starting from the
results of Andrews-Schilling-Warnaar.
