K-moduli of curves on a quadric surface and K3 surfaces

We show that the K-moduli spaces of log Fano pairs $({\mathbb{P}}^1\times {\mathbb{P}}^1,cC)$ where $C$ is a $(4,4)$-curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ complete intersection curves in ${\mathbb{P}}^3$. This, together with recent results by Laza-O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$-curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ and the Baily-Borel compactification of moduli of quartic hyperelliptic K3 surfaces.