Distinctness conjecture for coprime values of the divisor-sum ratio

Let $x,y\in\mathbb{N}$ be coprime and satisfy $x\ne y$. Distinctness conjecture. The values

$$\frac{\sigma(x)}{x}\quad\text{and}\quad\frac{\sigma(y)}{y}$$

are distinct. Here $\sigma$ is the sum-of-divisors function. Distinctness is known for squarefree inputs, but the conjecture remains open for arbitrary distinct coprime natural numbers and is related to the odd $k$-perfect number conjecture.