Independence of optimal degenerations from the weighting function

Let $(X,\Delta)$ be a log Fano pair and let $g,\bar{g}$ be functions satisfying the conditions in the source. Let $(Y,\Delta_Y,\xi_0)$ and $(\overline{Y},\Delta_{\overline{Y}},\bar{\xi}_0)$ be the $g$- and $\bar{g}$-optimal degenerations of $(X,\Delta)$, respectively. Independence conjecture. The underlying log Fano pairs are isomorphic:

$$(Y,\Delta_Y)\cong(\overline{Y},\Delta_{\overline{Y}}).$$

If true, this would show that the optimal degeneration of a log Fano pair is independent of the choice of admissible weighting function, even though the associated valuations, Reeb vectors, and weighted K-stability notions may vary. The conjecture is presented as an open question in the source; the precise hypotheses encoded by the referenced condition on $g$ should be checked.