Let $\sigma(x) = \sigma_{1}(x)$ denote the sum of the divisors of $x$, and let $$I(x) = \dfrac{\sigma(x)}{x}$$ be the abundancy index of $x$.
For example, $$\sigma(10) = 1 + 2 + 5 + 10 = 18$$ so that $$I(10) = \dfrac{\sigma(10}{10} = \dfrac{18}{10} = \dfrac{9}{5}.$$
Consider the equation $$I(x) = I(a)$$ where $a$ is fixed. If $x \neq a$, then $x$ is said to be a friend of $a$. We will then call $x$ and $a$ as friendly numbers. On the other hand, if $x = a$ whenever $$I(x) = I(a),$$ then $a$ is called a solitary number.
In this paper by Ward, necessary conditions for the existence of a friend of $10$ is derived. It is conjectured that $10$ is in fact solitary, although a proof appears to be very difficult.
My question in the present post is as follows:
What would be the mathematical consequences, if any, of proving that $10$ is solitary?