Anytime Convergence Rate of Gradient Descent

For an $L$-smooth convex function $f\colon\mathbb R^d\to\mathbb R$, gradient descent uses

$$x_{t+1}=x_t-\eta_t\nabla f(x_t),$$

where the stepsize sequence $(\eta_t)_{t\geq0}$ is fixed independently of the stopping time. What is the best anytime convergence rate, uniformly over all such $f$? In particular, do there exist a schedule and an exponent $\alpha>1$ such that, for every $L$-smooth convex $f$, every minimizer $x^\star$, and every $T\in\mathbb N$,

$$f(x_T)-f(x^\star)\leq \frac{L\|x_0-x^\star\|^2}{T^\alpha}?$$

The guarantee must concern $x_T$ itself, not $\min_{t\leq T}f(x_t)$, and must hold at every horizon.