Optimal first-order oracle complexity for heterogeneous Hölder-smooth sums

Let $\mathcal{J}$ be a finite index set, and for each $j\in\mathcal{J}$ let $f_j$ be a convex $(M_j,v_j)$-Hölder smooth function. Let $x_0$ be the initial point, $x^*$ an optimizer of $\sum_{j\in\mathcal{J}}f_j(x)$, and suppose

$$\frac{1}{2}\|x_0-x^*\|^2\leq R.$$

For a target accuracy $\epsilon>0$, the optimal first-order oracle complexity conjecture. The optimal first-order oracle complexity for minimizing $\sum_{j\in\mathcal{J}}f_j(x)$ is

$$\sum_{j\in\mathcal{J}} \left[\bar c_j|\mathcal{J}|^{\frac{1-3v_j}{1+3v_j}}\left(\frac{M_j}{\epsilon}\right)^{\frac{2}{1+3v_j}}R^{\frac{1+v_j}{1+3v_j}} \right],$$

where $\bar c_j$ depends only on universal constants and $v_j$. This conjecture proposes the optimal dependence on the number of summands, improving the match between the upper and lower complexity bounds; its resolution would characterize the optimal performance of first-order methods for heterogeneous sums.