Structured low-rank certificate conjecture for minimax gradient descent

Fix $N$, a $1$-smooth convex function $f$, an initial point $x_0$, a minimizer $x_\star$, and gradient-descent iterates $x_0,\ldots,x_N$ with constant stepsize $\alpha$. Write $f_i=f(x_i)$ and $g_i=\nabla f(x_i)$ for $i\in\{\star,0,1,\ldots,N\}$, and define

$$Q_{ij}=f_i-f_j-\langle g_j,x_i-x_j\rangle-\frac{1}{2}\|g_i-g_j\|^2.$$

Let $r=r(N)$ be the conjectured minimax rate. Structured certificate conjecture. There exist nonnegative multipliers $\lambda_{i,j}$ having the displayed banded form in the source, with positive vectors $a,b,c,d$, such that

$$\sum_{ij}\lambda_{ij}Q_{ij}=f_\star-f_N+r\left(\|x_0-x_\star\|^2-\left\lVert\left(x_0-\frac{1}{2r}\sum_{i=0}^Nc_ig_i\right)-x_\star\right\rVert^2\right).$$

This would give a formal performance-estimation proof of the strengthened minimax conjecture using only $O(N)$ parameters rather than general $O(N^2)$ certificate entries; its resolution is not supplied in the source.