Can a variational auto-encoder learn images composed of random noise at each pixel (each drawn from the same distribution)? Can a variational auto-encoder (VAE) learn images whose pixels have been generated from a Gaussian distribution (e.g. $N(0, 1)$), i.e. each pixel is a sample from $N(0, 1)$?

My gut feeling says no, because the VAE adds additional noise $\epsilon$ to the original image in the latent space, and if all images and pixels are random from the same distribution it would be impossible do decode/reconstruct into the particular input image. However, VAEs are a bit of a mystery to me internally. Any help would be appreciated!
VAE's try and model the distribution of your data. So it's not going to learn " images composed of random noise at each pixel" per se (though, if overfitting, it could remember them). But it would be very capable of learning the simple noise distribution from which you sampled your random pixels