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	   We show that the K-moduli spaces of log Fano pairs  			 		 $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ 		, 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 and 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.

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES