Optimized full cyclic coordinate descent has a superlinear numerical rate

Let $N$ denote the iteration count, and consider full fixed-step cyclic coordinate descent on a problem partitioned into $2$ blocks, allowing the update to use all past gradient information, with the step sizes optimized numerically. Optimized full cyclic coordinate descent conjecture. Full cyclic coordinate descent with optimized step sizes can achieve a faster convergence rate than

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

For $2$-block algorithms, the experiments suggest a rate of

$$\mathcal{O}(1/N^{1.47}).$$

This is a numerical conjecture based on performance-estimation-problem upper bounds and fits; a theoretical guarantee for the improved rate is not provided.