Conjectured Betti-diagram shape for edge ideals of generalized Andrásfai graph complements

Let $I({\rm GA}(t,k)')$ be the edge ideal of the complement of the generalized Andrásfai graph ${\rm GA}(t,k)$, and let $\beta_{i,i+j}(I({\rm GA}(t,k)'))$ denote its graded Betti numbers. Betti-diagram shape conjecture. For all $t,k\ge 3$,

$$\beta_{i,i+j}(I({\rm GA}(t,k)'))\neq 0$$

if and only if either $j\in\{2,\dots,t+1\}$ and $j-2\le i\le (j-1)(k-2)-1$, or $(i,j)=\bigl(t(k-2),t+2\bigr)$. The conjecture predicts the exact shape of the Betti diagrams of these edge ideals; the paper proves the complete shape for $t=3$ and obtains upper and lower bounds for $t\ge 4$, while the stated pattern is supported by computations.