Obviously, that blur is induced by post-processing while doing the conversion to a cube map.
One solution is to compensate for this blur by pre-enhancing the image -which is a usual technique used in FM-radio (pre-emphasis), Dolby, etc whenever loss of high-frequency content occurs at the end device.
Check this, I had the best results by upscaling the image 2X using Photoshop’s “Bicubic Sharper”, (without making the file larger, since the visual information is about the same), which makes a slightly sharper upscale than the neutral bicubic: https://jsfiddle.net/dlllb/1n23pr64/
Or, use a higher quality / higher res source image that isn’t just marginally clear.