Abstract: Doron Zeilberger has described a method for settling the $q$-case of the Macdonald-Morris root system constant term conjecture for any {\it specific} root system provided there is sufficient computer time, memory space and some luck. He illustrated the method by proving the $S(G_2)^{\vee}$ case. His method involves finding and solving a linear system of equations. We remove the element of luck by showing that it is always possible to construct a triangular system. We apply the method to the so far open $S(F_4)$ and $S(F_4)^{\vee}$ cases. A consequence of our triangularity result is that, in the equal parameter case, the Macdonald-Morris constant terms (for a fixed root system) form a $q$-hypergeometric sequence.