Unit fractions with shifted prime denominators

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1:p\text{ prime}\}$. The same conclusion holds for the set $\{p-h:p\text{ prime}\}$ for any $h\in \mathbb{Z}\mathrm{\backslash }\{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.