HKL conjecture for degree-8 del Pezzo pairs

Let ${\mathcal F}^\ast$ be the locally symmetric variety, let $\lambda$ be its Hodge line bundle, and for $s\in[0,1]\cap{\mathbb Q}$ define

$$\Delta(s)=\lambda+s(H_h+25H_u),\qquad {\mathcal F}(s):=\operatorname{Proj}(R({\mathcal F}^\ast,\Delta(s))).$$

Let $P^\ast$ denote the relevant open moduli space, let $\overline{P}^{GIT}$ be the GIT partial compactification of smooth del Pezzo pairs, and let $\overline{P}_{c}^K$ be the K-moduli space with parameter $c$. HKL conjecture for $P_8$. The section rings $R({\mathcal F}^\ast,\Delta(s))$ are finitely generated for all $s$; in particular, ${\mathcal F}(s)$ is a projective variety of dimension $18$. The varieties ${\mathcal F}(s)$ interpolate $P^\ast$ and $\overline{P}^{GIT}$, and there is an isomorphism

$$\overline{P}_{c}^K\cong {\mathcal F}(s)$$

under the transformation

$$s=s(c)=\frac{1-2c}{56c-4}.$$

In particular, the K-moduli walls coincide with the HKL walls $\{1/n\mid n=1,2,3,4,6,8,10,12,16,25,27,28,31\}$. This conjecture proposes that the HKL models recover the K-moduli wall crossing for degree-8 del Pezzo pairs and connect the period-theoretic and GIT compactifications; finite generation and the asserted identifications remain the substantive claims.