Usual compression standards are mostly adaptive, estimating statistical models of input from implicit prior distributions (e.g. the probability of A followed by B begins at p(A)p(B)), reasonable assumptions (e.g. scanlines in an image follow the same distribution), small and fixed tables and rules (e.g. the PNG filters): not only a low volume of data, but data that can only change as a major change of algorithm.

A neural network that models upscaling is, on the other hand, not only inconveniently big, but also completely explicit (inviting all sorts of tweaking and replacement) and adapted to a specific data set (further demanding specialized replacements for performance reasons).

Among the applications that are able to store and process the neural network, which is no small feat, I don't think many would be able to amortize the cost of a tailored neural network over a large, fixed set of very homogeneous images.

The imagenet64 model is over 21 MB: saving 21 MB over PNG size, at 4.29 vs 5.74 bpp (table 2a in the article), requires a set of more than 83 MB of perfectly imagenet64-like PNG images, which is a lot. Compressing with a custom upscaling model the image datasets used for neural network experiments, which are large and stable, is the most likely good application (with the side benefit of producing useful and interesting downscaled images for free in addition to compressing the originals).