Drori–Teboulle conjecture for the smooth convex gradient method

Let $f\in\mathcal{F}_{0,L}(\mathbb{R}^d)$ be a smooth convex function with a minimizer $x_*$, let $f_*=f(x_*)$, and let $R=\lVert x_0-x_*\rVert_2$. For $0\leq h\leq 2$, the gradient method with normalized step size $h/L$ generates iterates

$$x_{i+1}=x_i-\frac{h}{L}\nabla f(x_i).$$

Drori–Teboulle conjecture. Every sequence of iterates generated in this way satisfies

$$f(x_N)-f_*\leq \frac{LR^2}{2}\max\left(\frac{1}{2Nh+1},(1-h)^{2N}\right).$$

The conjecture gives the exact worst-case objective accuracy of the fixed-step gradient method. The source states that it was proved for $0\leq h\leq 1$, while the range $1<h<2$ remained open there; the bound is attained by two explicit one-dimensional functions.