Shimmering and separation of rendering at far distances when viewing at shallow angles with anisotropic filtering enabled

Enter flat world, have high render distance set (I have 32 chunks for my testing), enable anisotropic filtering at various levels (e.g. disabled, 2x, 4x, and 8x). Move camera around and move player up and down close to ground level, observing the horizon.

Expected result was that up to the end of the render distance there are no holes/separation/shimmering.

When testing, actual results showed there was high separation (holes showing through to the void) at 2x anisotropic filtering at far distances, similar but slightly reduced at 4x anisotropic, and only small amounts of shimmering at 8x anisotropic.