Abstract: We give a proof of the Macdonald-Morris root system conjecture for $F_4$ that draws on ideas from Zeilberger's recent proof of the $G_2^\vee$ case and Kadell's proof of the $q$-$BC_n$ case. Our proof depends on much computer computation. As in Zeilberger's proof the problem is reduced to solving a system of linear equations. A FORTRAN program generated the equations which were solved using the computer algebra package MAPLE.