Hello again Andrew,
Even if my diagnosis does not fit, the bleach treatment will kill whatever is attacking the leaves. I had something very similar attacking one of my EE's, & used this method to rid it & the soil of the infection. I sterilized a pair of scissors & removed the affected tissue deep enough into the unaffected growth to be sure it was not infected. I made sure not to cut the veins feeding the rest of the leaf. I then used a mix of 1 part chlorine bleach to 9 parts water with a sponge to systematically sterilize the full surface of the soil. I left the plant unwatered for 1 day afterwards to allow the chlorine to evaporate, & saw no more problems. I said black mold because I found the problem listed as a common problem for EE's when using misting or high humidity in closed rooms. I have since moved my EE's outdoors since they like the higher intensity light & the cooler breezes (relative since I am in South America, EE's do not like "cool" at all). I do not know the affects of neem oil, but if you are careful not to dench the plant with chlorine bleach this will work. I have used this method when mold attacks the soil of my plants several times, & you can see my collection on ATP.
Noel