Betti-table symmetry conjecture for graphs formed from glued four-cycles

Let $G$ be a graph formed by gluing $k$ copies of the four-cycle $C_{4}$ together along one edge. Write $b_{i,j}(\overline{G})$ for the graded Betti numbers of its graph curve. Betti-table symmetry conjecture. Then

$$b_{2,4}(\overline{G})=b_{1,2}(\overline{G})-1,$$ $$b_{i,i+1}(\overline{G})=b_{i+2,i+4}(\overline{G}),\qquad i\geq 2,$$

and

$$b_{k,k+2}(\overline{G})=\left(b_{k+1,k+3}(\overline{G})\right)^{2}-1.$$

These experimentally observed identities describe patterns between the quadratic and cubic strands for this family of graph curves; the source gives them as conjectural future work, with no resolution stated.