Polytope conjecture for achievable derivatives in the spin-spin system

Let $\Delta^3$ denote the probability simplex, let $\mathsf{derv}(\lambda)$ be the set of achievable derivatives at $\lambda$, let $S_4$ be the permutation group on four elements, and let $L_P$ denote the operator associated with $P\in S_4$. For a set of vectors, write $\operatorname{conv}$ for its convex hull. Spin-spin polytope conjecture. For every $\lambda\in\Delta^3$ it holds that

$$\operatorname{conv}(\mathsf{derv}(\lambda))=\operatorname{conv}(\{-L_P\lambda:P\in S_4\})=:\mathcal P(\lambda).$$

The conjecture proposes an exact polytope description of the convex hull of achievable derivatives for the two-qubit spin-spin system with a single Lindblad term $V=\sigma_-\otimes\mathds1$. The paper supports it with a partial proof and numerical evidence, while establishing a slightly larger upper bound and deriving an optimal cooling procedure separately.