I believe that the white fungi are plenty good for decomposition and exist in parallel with worms in almost any sort of compost bin. They are very common in animal manure/straw mixtures. Surprisingly I used the juice every few days from the 3wk fermented bucket on a fresh built 4x4x4 compost pile of my garden waste and have never seen so much of this white fungi so immediately, and have also never seen a pile diminish so quickly. It was dropping inches every day in height. Generally I see inches/week on these sized piles.

My plan with the full bokashi bucket is to add an equal part to twice as much of a mixture of paper scraps/egg cartons/peat moss/coco coir/leaf duff/leaves/whatever I have around. I will also add my mixture of minerals that I add to my soil amendments, including basalt, bentonite, oyster shell powder, and glacial rock dust. To this mixture I will be experimenting with the quantity needed of oyster shell powder to balance the pH as I have read a fully pickled bucket will generally gives pH readings as low as 3.0, which is probably quite a bit lower than any worms would prefer.

Hope this helps and if anyone still cares, feel free to respond.