Rational as Prime Egyptian Fractions

Conjecture
For any positive rational $r$ there is a finite set of prime numbers $P_r^{+}$ such that $\sum _{p\in P_r^{+}}\frac{1}{p+1}=r$
For instance $\frac{3}{14}=\frac{1}{5+1}+\frac{1}{23+1}+\frac{1}{167+1}$

Can you verify this conjecture for $\frac{25}{61}$?

Answer format : $p_1,p_2,\dots ,p_n\text{ if }\frac{25}{61}=\frac{1}{p_1+1}+\frac{1}{p_2+1}+\cdots +\frac{1}{p_n+1}$
As there are generally more than one solution, give the 1st one in lexicographic order
Limit your search to prime numbers $<100000$

[My timing: 35 sec]