Can the sum of finitely many inverses of distinct odd integers $\geq 3$ be 1? Is there a positive number $n$ of distinct odd integers $z_1,z_2, \ldots, z_n \geq 3$ such that $\frac{1}{z_1} + \frac{1}{z_2} + \cdots + \frac{1}{z_n} = 1$?

In 1954, it was shown by Stewart and Breusch (independently) that if $\frac {p}{q} >0$ and $q$ is odd, then it can be written as the sum of finitely many reciprocals of odd numbers.
As a specific example,
$$1=\frac {1}{3} + \frac {1}{5} + \frac {1}{7} + \frac {1}{9}  + \frac {1}{15} + \frac {1}{21} + \frac {1}{27} + \frac {1}{35} + \frac {1}{63} + \frac {1}{105} + \frac {1}{135}$$
