Obláth’s prime divisor condition

For every pair of natural numbers $n$ and $q$, if $n>2$, $q$ is prime, $q$ divides $n+1$, and the remainder upon dividing $q$ by $4$ is $3$, then there exist natural numbers $x,y,z$ such that $1\le x$, $x<y$, $y<z$, and $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$, where the natural numbers in this equation are regarded as rational numbers and all divisions are rational divisions. The hypotheses exclude $n=0,1,2$ and $q=0,1$, and the inequalities require all three denominators to be positive and pairwise distinct, so no division by zero occurs in the asserted equation.