Distinctness conjecture for the values of the divisor-sum ratio

Let $\sigma(x)$ denote the sum-of-divisors function for $x\in\mathbb{N}$. Distinctness conjecture. All values of

$$\frac{\sigma(x)}{x^2}$$

for $x\in\mathbb{N}$ are pairwise distinct. No equal values are known for different natural numbers, although distinctness is known for different coprime numbers. The conjecture concerns whether the function $x\mapsto \sigma(x)/x^2$ is injective on $\mathbb{N}$.