The modified Moore–Mariño conjecture for the ${\rm U}(3)$ series

Let $X$ be a four-manifold of ${\rm U}(3)$-simple type, let $\{K_i\}$ be its Kronheimer–Mrowka basic classes, and let $\mathfrak s_i$ be chosen so that the basic class associated to $\mathfrak s_i$ is $K_i$. The modified Moore–Mariño conjecture. The ${\rm U}(3)$ series has the form

$$\widehat{{\rm D}}_{X,w}({\rm e}^{\Gamma_{(2)}+\Lambda_{(3)}})={\rm e}^{\frac{Q(\Gamma)}{2}-Q(\Lambda)}\sum_{i,j}c_{ij}\zeta^{-w\cdot\left(\frac{K_i-K_j}{2}\right)}{\rm e}^{\frac{\sqrt 3}{2}(K_i+K_j)\cdot\Gamma+\frac{\sqrt 3}{2}{\bf i}(K_i-K_j)\cdot\Lambda},$$

where

$$c_{i,j}=2^{\chi+\frac{3}{2}\sigma+\frac{1}{2}K_i\cdot K_j}3^{2+\frac{7}{4}\chi+\frac{11}{4}\sigma}SW_X(\mathfrak s_i)SW_X(\mathfrak s_j).$$

The paper states that its calculations agree with this modified conjecture and can be used to determine previously undetermined constants, but does not claim a complete proof.