How to download via Chrome on Nexus 7 with Jelly Bean 4.2.2
I tried to download the link using Chrome for Android SEVERAL times, but it always disappears. Like I've been telling myself before, Android for Chrome is not mature enough and lacking in (so many) features.

I would suggest installing a 3rd-party browser like Dolphin or Boat, which are both highly-customizable. I downloaded the MP3 file successfully using Boat (which is my main browser), and it appears in my Download folder.