An Egyptian fraction is a finite sum of distinct unit fractions, such as$$\frac{43}{48} = \frac{1}{2}+\frac{1}{3}+\frac{1}{16}.$$
Does there exist a number in the range $(0.5, 1)$ that when written as an Egyptian fraction, none of the representations with the minimum length include $\frac{1}{2}$?
For example, I proved computationally (by checking all examples in my other question) that if you can write a number in the range $(0.5, 1)$ as $\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$ you can also write it as $\frac{1}{2}+\frac{1}{s_1}+\frac{1}{s_2}$. I'm wondering if it is a general rule for any minimum length.
