iTunes stopped updating my podcast

I can't see any reason for this. The only problem in the feed is the use of dodgy characters in the frequently used phrase

If youâ\x80\x99re reading

However this applies to earlier episodes than 55; the Store evidently hasn't updated since April though when subscribing episodes up to 63 show. It does happen sometimes that the Store gets stuck for a few weeks for no apparent reason. I can only suggest contacting Support, though I can't promise they will be of any help. Go to https://itunespodcasts.zendesk.com/hc/en-us/requests/new and click the disclosure triangle at the right of the 'Choose a topic' field - choose an appropriate topic and proceed from there.

As you are with Libsyn you may be able to get some help from them.

A 502 error is 'bad gateway' - 'A server (not necessarily a Web server) is acting as a gateway or proxy to fulfil the request by the client (e.g. your Web browser or our CheckUpDown robot) to access the requested URL. This server received an invalid response from an upstream server it accessed to fulfil the request.' (See http://www.checkupdown.com/status/E502.html ).

I don't see any issues in a browser or using curl in Terminal (which would show any redirects). The only explanation I can think of is that the server is not responding to requests specifically from the iTunes Store. I've seen cases where the Store couldn't access feeds where everything else could, though i've never seen any explanation for this. I can only suggest that you take this up with the people running your server - either that, or try hosting the feed on a different server. Or, since you are hosting the media files on Libsyn, I would have thought that they would provide a feed which might be preferable to use.

I'm glad to hear they were able to help you.