The centre-generation conjecture for weak generalized Satake subalgebras

Let $(X,\tau)\in\operatorname{WSat}(A)$, let $\mathfrak k$ be the associated Lie algebra, and let $\mathfrak z$ denote its centre. Define

$$\mathcal J_{\rm even}:=\{\boldsymbol j\in\mathcal J\mid\text{the coefficient of }\alpha_k\text{ in }\alpha_{\boldsymbol j}\text{ is even for all }k\in I\setminus X\}.$$

Let $\mathfrak k(i)_2$ be the indicated degree-two component of the subalgebra associated with the unique relevant $i$.

Centre-generation conjecture. A single element of

$$\bigoplus_{\boldsymbol j\in\mathcal J_{\rm even}}\mathbb C b_{\boldsymbol j}\subset\mathfrak k(i)_2$$

generates $\mathfrak z$.

The assertion describes the centre of the non-reductive Lie algebras associated with weak generalized Satake diagrams. It is stated without a resolution in the supplied text.