[matjet]

Gaussian blur is essentially acting as a low pass filter. An IR filter does not strictly destroy information in the filtered spectrum components, but does attenuate their power.

Given a perfect blurred image, reconstruction is possible - however due to the attenuation, these high frequency components are ~sensitive~.

Apart from quantisation effects [you mentioned which limits perfect de-convolution], adding a little AW Gaussian noise(such as taking a photo of the image from across the room) after the kernel is applied obliterates high frequency features.

Recovery when noise is low (plus known glyphs) is why you should not use Gaussian blur followed by print screen to redact documents. Inability to recover when there are artifacts and noise is [part of] why cameras cannot just set a fixed focus [at whatever distance] and deconvolve with the aperture [estimated width at each pixel] to deblur everything that was out of focus.

TLDR for readers, It is unlikely to recover sufficient detail via de-convolution here.