Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know):

Are there (known) lower and upper bounds to the following arithmetic / number-theoretic expression:

$$\frac{I(x^2)}{I(x)} = \frac{\frac{\sigma_1(x^2)}{x^2}}{\frac{\sigma_1(x)}{x}}$$

where $x \in \mathbb{N}$, $\sigma_1(x)$ is the sum of the divisors of $x$ and $I(x) = \frac{\sigma_1(x)}{x}$ is the abundancy index of $x$?

(Note that a trivial lower bound is $1$ since $x \mid x^2$ implies $I(x) \leq I(x^2)$.)

I would highly appreciate it if somebody will be able to point me to relevant references in the existing literature.

Solutions Collecting From Web of "Are there (known) bounds to the following arithmetic / number-theoretic expression?"

Will Jagy has posted an answer to a closely related MSE question here.

Per Will Jagy’s answer in the linked MSE question (and a subsequent comment by Erick Wong), we have the conjectured (sharp?) bounds

$$1 \leq \frac{I(x^2)}{I(x)} \leq \prod_{p}{\frac{p^2 + p + 1}{p^2 + p}} = \frac{\zeta(2)}{\zeta(3)} \approx 1.3684327776\ldots$$

I would still be interested in an (improved) lower bound for $I(x^2)/I(x)$ when $\omega(x) \geq 3$, where $\omega(x)$ is the number of distinct prime factors of $x$.