Construction of the projective plane $PG(2,q^2)$ from the unitary group $PSU(3,q)$

In 2013, the first and the second author of this paper described a construction of the projective plane $\text{PG}(2,q^2)$ from the unitary group $\text{PSU}(3,q)$, for $q=3,4,5,7$. The construction is obtained by using a computer. In the same paper, it is conjectured that in a similar way one can construct the projective plane $\text{PG}(2,q^2)$ from the unitary group $\text{PSU}(3,q)$, for every prime power $q$. In this paper, we give a construction of a Desarguesian projective plane from a unitary group that confirms this conjecture.