The relative-invariant hierarchy conjecture for toric Fano surfaces

Let $X$ be a toric Fano surface. Let $D$ be a toric invariant divisor, and let $D^+$ and $D^-$ be disjoint toric invariant divisors. Consider the suitably defined generating series of invariants on moduli spaces of maps to $X$ relative to these divisors. Relative-invariant hierarchy conjecture. The generating series relative to $D$ is a $\tau$-function of the KP hierarchy, while the generating series relative to $D^+$ and $D^-$ gives rise to a sequence of $\tau$-functions of the $2$-Toda hierarchy. This is proposed based on the preceding examples; the source does not state a proof or resolution.