General-$k$ upper-bound conjecture for temporal Wasserstein forecasting

Let $k\ge1$. On a regular problem with an equispaced dense design, consider a degree-$k$ temporal local-polynomial forecaster on the tangent bundle, obtained by geodesic or barycentric regression of the snapshots with sample splitting. Let $\varepsilon$ denote the temporal regularity scale, $h$ the forecast horizon, $H$ the estimation window, $M$ the total sample-size parameter, $L$ the temporal-design parameter, and $\gamma_d$ the Wasserstein empirical-convergence exponent in dimension $d$. The order-$k$ Otto--Taylor remainder gives bias of order $\varepsilon(h+H)^{k+1}$, while the variance is of order $(MH/L)^{-\gamma_d}$. General-$k$ upper-bound conjecture. The forecaster attains these bias and variance bounds, and hence meets the lower bound with unified exponent

$$M^{-(k+1)/(d(k+1)+1)}.$$

The construction and constant are currently established only in the location channel for all $k$ and end-to-end for $k=0$; the general unconditional result is reduced to a curvature-stability estimate and an optimal-transport map-estimation rate, both known unconditionally at $k=0$ and on flat submodels. Thus the unconditional upper bound for $k\ge1$ remains open.