Basic g-composable schedules capture minimax-optimal gradient-norm schedules

For each positive integer $n$, let $h\in\mathbb{R}^n$ be a fixed-step gradient-descent schedule, and let $x_n$ be the resulting iterate. Let $\mathfrak{F}_{1,\delta}$ be the set of all problem instances $(f,x_0)$ in which $f$ is $1$-smooth and convex and the initialization satisfies

$$f(x_0)-f(x_\star)\leq\delta.$$

A schedule is basic if it is built from the empty schedule using the paper's composition operations, and it is $g$-composable if it has the paper's gradient-norm convergence guarantee.

Basic g-composable schedule conjecture. For each $n$, every minimax-optimal stepsize schedule solving

$$\min_{h\in\mathbb{R}^n}\;\max_{(f,x_0)\in\mathfrak{F}_{1,\delta}}\frac{1}{2}\|\nabla f(x_n)\|^2$$

is basic and $g$-composable.

This is presented as complementary to the conjecture about OBS-F schedules and is motivated by the exact correspondence between basic $f$-composable and basic $g$-composable schedules. The supplied text gives no resolution status.