Optimized full inexact gradient descent has a superlinear numerical rate for residual gradient minimization

Let $N$ denote the iteration count and let the performance criterion be the minimum squared gradient norm, $\min_{k\in\{1,\ldots,N\}}\|\nabla f(x_k)\|^2$, for minimizing a smooth convex function with relative inexactness level $\epsilon$. Consider full inexact gradient descent, whose updates may use all past inexact gradient information, with numerically optimized step sizes. Optimized full inexact gradient descent conjecture. Full inexact gradient descent with optimized step sizes can achieve a faster convergence rate than

$$\mathcal{O}(1/N)$$

for minimizing the squared gradient norm. The reported rates for $\epsilon=0.1,0.3,0.5$ are respectively $\mathcal{O}(1/N^{1.384})$, $\mathcal{O}(1/N^{1.342})$, and $\mathcal{O}(1/N^{1.190})$. These are numerical fits, and the source does not provide a theoretical proof of the rates.