In Hügelkultur, isn't there a lack (or a fast depletion) of oxygen for fungal development ?

Without further study I would say the influx of newly oxygenated water would counter any depleted pockets , and they fact they are in free draining above ground, even though covered with soil, would allow some oxygen through osmosis ... Just my thoughts .... Dave oxoxox

Aside from the water bringing oxygen, there are also plant roots that create channels to and through the wood. Mice, voles, moles and a plethora of insect life will tunnels as well. This causes an air exchange for the mound. Remember that many species of ants and termites can live many meters below the ground because of the way they build the tunnels in the colony. In a similar way, once the bed is established, it will create the optimal environment to grow. Fungus is a part of the system which will help determine how long the wood lasts in the bed. You may notice that the bed doesn't perform uniformly, as there are many factors that influence the performance such as...

Climate
water
temperature
type of wood
size of wood pieces
decomposition level of wood before burying
health of soil prior to construction
aspect
rainfall amounts and frequency
what you grow in the mound
depth of wood
percentage of wood compared to overall mound volume
Permeability of soil
Permeability of wood
soil life present
Total biomass
mineral profile
nutrient availability
pH

In my experiments so far, hugelkulture is best practiced as a "go big or go home" type of thing. For the effort involved, you want the most out of your bed. Build them bigger and wider than you'd think you'd want them. It'll pay off in the long run as they will shrink to some degree as they settle and rot. I kind of thought of it like I was building a fire in ground. The idea is to get small things to rot and then that rot spreads to the bigger things.You'll want this to be a slow process so that you get the longest life you can out of the bed. Building it for optimal fungal development would rot the wood so fast that it would defeat the effort. You want the fungus alive and healthy but not ravaging the wood. That wood is for holding water and feeding the plants. The fungus is there to turn the wood into nutrients and a water basin SLOOOWWWWWWLY.
Dig the hole then add small wood, some green stuff like leaves or grass, big wood, more small wood, soil, compost, mulch and then plants. I would think that there's so much bio-activity in the soil to get the whole thing going in pretty quick time.

The only way to mess it up is to bury the wood so deep that roots, water and air are too far away to get the mound "alive". Just guessing, but I don't think anyone has bothered to dig a bed that deep. It would have to be pretty deep down in the subsoil. Maybe 20 feet?

Don't worry.... it'll work.
Best wishes

It doesn't need much to survive. I have test tube cultures and liquid cultures that are going on their seco d year and still very alive

And in some ways, you are trying to slow/control the decay. You don't want the fungi to over thrive and release everything too fast.

I have five hugelbeds completing their second year of existence. If anything, the fungi were visibly more active this year than last. Based on the number of mushrooms arising all over the beds, I'm not worried about lack of oxygen. I covered the beds with approximately six inches (15 cm) of soil originally and mulched with about 2cm of wood chips. Got voles in some places over last winter, some air pockets collapsed opening the inside of the mound to the atmosphere and plants are rooting well in most places. I would concur that growth in my beds is not uniform but some areas are growing more vigorously than others. My beds are just over a meter to 125 cm tall and nearly 2m wide. Lots of large rounds, a few stumps, split partially rotted firewood, and sticks from tree trimming make up the cores of the beds. I don't see that the decomposers are lacking oxygen to do their job.