Conjectured identities among shifted G-determinants

For integers $\alpha,\beta,\gamma,\delta$, define

$$G_{\alpha,\beta,\gamma,\delta}(n):=\det_{0\leq i,j\leq n-1}\left(2^{i+\beta}\binom{i+4j+\gamma}{4j+\alpha}+\binom{-i+4j+\delta}{4j+\alpha}\right).$$

Shifted G-determinant identities. For all $n\geq3$, the three chains of identities displayed in the source hold:

$$G_{3,0,3,3}(n)=\frac23G_{0,1,-2,-4}(n+1)=-\frac1{672}G_{1,3,-2,-8}(n+1)=\frac1{63}G_{5,4,3,-5}(n)=\frac4{1002001}G_{6,6,3,-9}(n)=-\frac85G_{9,5,8,-2}(n-1),$$ $$G_{1,1,0,-2}(n)=-\frac1{49}G_{2,3,0,-6}(n)=-\frac27G_{6,4,5,-3}(n-1)=-\frac4{5577}G_{7,6,5,-7}(n-1),$$ $$G_{2,1,2,0}(n)=2G_{7,4,7,-1}(n-1).$$

These identities relate experimentally identified determinant evaluations and are stated conjecturally in the source. The surrounding proposition establishes some related block-structure identities, but the displayed chains themselves are left unproved.