I would put him in the fridge to be safe, my little girl Peppa had fungus appear on her gills almost overnight in the same way. I gave her a salt Bath but hesitated about putting her in the fridge as to be honest was a little weary of it. I have regretted that decision as she has now completely lost the gills on one side and went through a lot of stress.
Then my little boy Carlos got the fungus so that was it straight into the fridge. His fungus didn't spread at all but halted in it's tracks. And with salt bath treatments as now completely gone.
I would recommend you salt bath him twice a day if possible for 10 mins at a time (no longer then 15 as it is harmful), until the fungus appears to have gone.
If you are going to fridge him (which I strongly suggest) and also salt bath him the best thing do to is to follow these simple steps.
Insure your fridge temp is above 4 degrees celsius as under this can be harmful to your axi. Test by placing a cup of water with a thermometer in it at the back of your fridge - leave it for 15 mins then come back to check. I would recommend a temp in between 5-7 degrees.
Place your axi into at least a 2-3ltr container of dechlorinated water just enough to cover him completely, place a lid on top and wrap the container with a teatowel to prevent fluctuations in temp and guard against the light. Then place him into the fridge, this will allow the water to come down in temperature slowly.
Place a spare container the same size in the fridge of dechlorinated water - this will be used for salt baths. Also place a bottle of dechlorinated water in the fridge for water changes.
In the morning remove your axi and spare water container from the fridge, mix 2 teaspoons of salt for every ltr of water into the other container of water. Using a net remove your axi from its container and bath for ten mins. Your axi probably wont like this and will wriggle around a bit, you can place a lid on top during the bath and eventually he should calm down and accept his situation. Then after the bath put your axi back in his container and back into the fridge. After he is safely away clean the bathing container using a mix of water and baking soda. Rinse and Refill container with water and dechlorinate. Place back into fridge for the evening salt bath.
For the evening bath follow the same process as morning bath, however when the little one is in the bath drain the water from his original container. Clean with water and baking soda. Rinse then pour water from the bottle kept in the fridge into the clean container. Follow the rest as per the morning, and refill the bottle with dechlorinated water and back into the fridge.
This process is better as it prevents temperature fluctuations form stressing the axi. When you need to return your axi to its tank make sure you leave the container out to slowly come to the tank temperature before putting him back in the tank as this will limit any stress.
Don't stress if your axi doesn't eat while he is in the fridge, as his metabolism will slow at lower temps. If he starts to look a little skinny, try pulling him out of the fridge for a day to raise his temp so he may be little more inclined to eat. You can also try feeding your axi just before his evening bath every night just to see if he will eat.
Treatment can take anywhere from 1 to 3 weeks + depending. I also like to use Stress Coat water conditioner while they are going through treatment.
While you axi is in the fridge I would take the time to look over your tank to see what could be the cause. If your ammonia, nitrite, nitrate and PH levels are fine then you could look at possible other causes such as, water temp, new plants, gravel in tank, feeder fish etc.
Sorry to overload you with info, I just know I had to flick between so many posts to answer my questions for my little ones. I hope this helps you.