Not really, Ray.
I have this small white umbrella looking mushrooms in a few pots.
Some of these pots contain rather old potting mix but I do have these mushrooms growing out of newly repotted (barely a few months old) pots also. I suppose the pores are all flying around in the air and find their way to random spots in my collection.
Bark, whether new or old, is dead plant material and as long as there is moisture, it will be their food source.
I don't know the exact biology behind this particular fungus so I don't know what they prefer, but take shitake mushroom cultivation for an example.
People who farm this fungus drill holes on the "fresh dead" tree related to oak (there are about 8 species of trees that are used for cultivating Shittake mushroom in Korea, Japan and China), not long dead half decomposed one, and plant the spores in the holes.
The trees to be used for mushroom farming are harvest in the middle of winter and then dried for minimum of one month.
Fungus are among the first decompoers of dead organic materials. So, fresh bark is a great source of food for them.
By the way, my plants with the little mushroom growing out of the pots are doing great.
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