The semi-algebraicity conjecture for K-semistable domains

Let $X$ be a normal projective variety, and let $D_j$ for $1\leq j\leq k$ be effective Weil divisors on $X$. The notation $\operatorname{Kss}(X,\sum_{j=1}^k D_j)$ denotes the K-semistable domain associated with these divisors. Semi-algebraicity conjecture. The set

$$\operatorname{Kss}\left(X,\sum_{j=1}^kD_j\right)$$

is a semi-algebraic set. A semi-algebraic description of the K-semistable domain is identified as an important ingredient in extending the wall-crossing results without the volume condition $(\clubsuit)$; its resolution would support the conjectural volume-free chamber decomposition.