Whether or not Border Mip Maps fixes this depends on the content of the image. If the texture you’re using has color all the way to the edges, as it should for a horizontally repeating texture like this example, the 1x1 will end up very similar to the mip map w/o the Border Mip Maps enabled. So in this case it will not fix the problem.
For example, here’s a similar noisy glow texture like what’s being used above, and the 1x1 mip map with and without the Border Mip Maps enabled.
They look almost identical, and in this case the one with it enabled is slightly brighter!
But if you add a 1 pixel border around the entire image it will make the 1x1 mip disappear.
But this just adds a new issue, which is there’s an obvious black seam when the texture repeats, and it’s a line that “grows” with the mip map as it’s always 2 texels wide.
That’s why I didn’t mention it as a fix for this case. It works for more punctual fx textures, but not for any texture that needs to wrap.