HPV was "cured" (cleared) by the human body itself within 2 months in 5% of the control subjects (n=20) and in 88% of the test group (n=41).
I think they were concerned that various tree bark needed to grow the fungus could itself influence the outcome and wanted to "control" for any effects of the bark. Part IV says "LS grown on the same barks as TV or GL was chosen as a control since no anti-HPV property has yet been reported with this mycelium and because the influence of barks should be excluded."
Perhaps there are more studies out there with different "controls" that would add to this research to help rule out any influence from unknown properties of Laetiporus sulphureus.