Sounds like this might be your area of expertise. For the rest of us, take a shoebox. How much ballpark extra weight we talkin’ to have a livable planet? (Maybe the mushrooms would be ~2x as heavy as standard shoeboxes for example, to meet existing spec.)
Or how about for the glasses box they show on the site in OP, or a plastic sleeve like Americans sell Oreo cookies in. Anybody have any guesses?
I've done some experiments with mycelium as a construction material, but I'm hardly an expert. Mycelium weighs anywhere between 50 and 950kg/m3. Usually you won't have mycelium as thin as cardboard, because you want use mycelium as a 3d buffer, replacing styrofoam. EPS (styrofoam) has densities of 15-30kg/m3. So while it's more sustainable it's also heavier.