Strongly convex gradient-norm worst-case conjecture

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

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

Strongly convex gradient-norm conjecture. Every sequence of iterates generated in this way satisfies

$$\lVert\nabla f(x_N)\rVert_2\leq LR\max\left(\frac{\kappa}{(\kappa-1)+(1-\kappa h)^{-N}},\lvert1-h\rvert^N\right).$$

The conjecture predicts the exact worst-case gradient norm through one-dimensional piecewise quadratic functions, analogously to the objective-value results. It is based on numerical experiments in both the smooth convex and smooth strongly convex settings, and no proof or resolution is given in the source.