I am sorry to hear about this concern for Jake.
A few things to cover:
1. The fecal test for worms. You need to find out from your vet if the fecal test was sent out to a lab or done in the hospital. Sent out to the lab is more accurate. Also, ask if the fecal test included a giardia elisa. It is a more sensitive test for diagnosing giardia, which is hard to find.
2. The elimination diet. I do not consider a diet appropriate for food allergy elimination unless it is a prescription food. Over-the-counter foods will not be sufficient. Also, grain-free is overrated severely. Most food allergies are due to the protein not the carbohydrate.
My suggestion: Ask the vet for Royal Canin HP Hypoallergenic formula. It is well-balanced soy based hydrolyzed food. Excellent for ruling-out food allergies.
3. Start metronidazole. It is a great antibitoic for helping with intestinal problems and intermittent diarrhea. Also, quite safe at lower doses for long periods of time.
4. Stop all treats. Everything!
5. If none of the above having any impact, than possibly talk to the vet about doing a "fecal culture" test. This tests for other tests of bacterial problems in the stool.
6. Another test to consider for chronic diarrhea is EPI, Exocrine pancreatic Insufficiency, where the pancreas is not producing proper enzymes for digestion. It is somewhat common and frequently undiagnosed because vets forget about it as a differential for the diarrhea.
Hope that information helps