Weyl chamber and stable-base-locus decomposition conjecture for $X^3_8$ and $X^5_9$

Let $X=X^5_9$ or $X=X^3_8$. Write $\overline{\operatorname{Eff}}(X)$ for the pseudoeffective cone of divisors, $\operatorname{WCD}(\overline{\operatorname{Eff}}(X))$ for its Weyl chamber decomposition, and $\operatorname{SBLD}(\overline{\operatorname{Eff}}(X))$ for its stable-base-locus decomposition. A rational contraction is a birational map $f:X\dasharrow Y$ of the type considered in the paper, and $\operatorname{ex}(f)$ denotes its exceptional contribution. The Weyl chamber decomposition conjecture.

$$\operatorname{WCD}(\overline{\operatorname{Eff}}(X))=\operatorname{SBLD}(\overline{\operatorname{Eff}}(X)).$$

2. Each Weyl chamber has the form

$$f^*\operatorname{Nef}(Y)*\operatorname{ex}(f),$$

where $f:X\dasharrow Y$ is a rational contraction.

This conjecture concerns the first non-Mori-dream cases where the pseudoeffective cone has only a negative part. It proposes that the Weyl chamber decomposition captures the stable-base-locus decomposition and that every chamber arises from a rational contraction.