Sign conjecture for fundamental identities of generalised spherical minors

Let $(G,\widehat G)$ be a spherical pair with the notation of Theorem 3.1: for each $b\in\mathcal B$, let $U_b\subseteq W$ be the associated set and let $\epsilon_b\in\{\pm1\}$ be the sign appearing in the fundamental identity for generalised spherical minors.

Sign conjecture. For every $b\in\mathcal B$,

$$\epsilon_b=(-1)^{|U_b|}.$$

The conjecture predicts a uniform description of the sign in the fundamental identities. The paper states that a pair-independent description of the associated data is generally missing, and presents this sign formula as an open conjecture.