Unique q-series decomposition conjecture for Mordell–Borel integrals

Let $p$ be an odd integer and let $J^{(A,p)}_j(t)$, $J^{(B,p)}_j(t)$, $J^{(C,p)}_j(t)$, and $J^{(D,p)}_j(t)$ denote the vectors of Mordell–Borel integrals defined in the source. Write $q$ and $\widetilde{q}$ for the parameters associated with the $t>0$ and analytically continued regions, respectively. Let $A$, $B$, $C$, and $D$ be the mixing matrices from the unary-side decompositions, and let $\Phi_p^{(a)}(q)^{\vee}$ be $q$-series for $a=1,\ldots,p-1$. Unique decomposition conjecture. For all odd $p$, the Mordell–Borel integrals at $t>0$ decompose uniquely into $q$-series and $\widetilde{q}$-series according to the displayed identities in the conjecture, and the dual of $\Psi_p^{(a)}(q)$ is

$$\Psi_p^{(a)}(q)^{\vee}=q^{-\frac{a^2}{4p}}\Phi_p^{(a)}(q)^{\vee},\qquad a=1,\ldots,p-1.$$

This conjecture asserts that the algebraic decomposition relations on the unary side are preserved by unique continuation to $t>0$, identifying the dual $q$-series associated with the false theta functions.