The prime frontier at one modulo twenty-four

The following assertions are equivalent: (i) for every natural number $n>2$, there exist natural numbers $x,y,z$ such that $1\le x<y<z$ and $\frac{4}{n}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$; (ii) for every natural number $p$ that is prime and has remainder $1$ upon division by $24$, there exist natural numbers $x,y,z$ such that $1\le x<y<z$ and $\frac{4}{p}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$. All fractions and equalities are interpreted in the rational numbers, with the natural numbers embedded into them. The witnesses may depend on $n$ or $p$. Although natural numbers include $0$, the denominator inequalities require three positive, pairwise distinct denominators in increasing order, and neither assertion requires a decomposition for $0$, $1$, or $2$.