Uniqueness of the eta partner for an image of the zeta map

Let $a$ and $b$ be relatively prime positive integers. Let $\mathfrak{D}_{a,b}$ be the set of $(a,b)$-Dyck paths, and let

$$\mathfrak{Z}=\{(\zeta(P),\eta(P)):P\in\mathfrak{D}_{a,b}\}\subseteq\mathfrak{D}_{a,b}\times\mathfrak{D}_{a,b}$$

be the image of the pair $(\zeta,\eta)$. Uniqueness conjecture for eta partners. For every $(a,b)$-Dyck path $Q$ in the image of $\zeta$, there exists at most one $(a,b)$-Dyck path $R$ such that $(Q,R)\in\mathfrak{Z}$. Together with the preceding theorem in the source, this would imply bijectivity of the zeta map; the source gives no resolution status.