The multidimensional BFECC interpolation accuracy conjecture

Let $d$ be a positive integer, and consider a uniform rectangular grid in $d$ dimensions with mesh sizes

$$\Delta x_1,\Delta x_2,\cdots,\Delta x_d.$$

Let the grid be shifted along any vector, and apply BFECC interpolation from the original grid to the shifted grid. BFECC interpolation accuracy conjecture. BFECC interpolation is third-order accurate. In particular, if the shifted grid points are located at centroids of the original grid cells, then BFECC interpolation is fourth-order accurate. The result is established in the paper for one and two dimensions, with numerical experiments indicating that it also holds in three dimensions; the conjecture extends this accuracy statement to every positive dimension.