On odd-spin $A_{1}^{(1)}$-string functions, cross-spin identities, and mock theta conjecture-like identities

Determining the explicit forms and modularity for string functions and branching coefficients for Kac--Moody algebras after Kac, Peterson, and Wakimoto is a long-standing, yet wide-open, problem and recently a connection has been made between positive admissible-level $A_{1}^{(1)}$-string functions and Ramanujan's mock theta functions. In this paper we obtain the polar-finite decomposition for the admissible-level $A_{1}^{(1)}$ character of odd spin, and we also find new mock theta conjecture-like identities for the odd-spin, $2/3$-level and $2/5$-level $A_{1}^{(1)}$-string functions.