This is an update of my original response which was incorrect.

The conjecture is false. I will show that any $X$ in the conjecture has density at most

$$ 0.4226496 < 1-\frac{1}{\sqrt{3}}= 0.4226497...$$

Let $Y$ be the set of odd numbers whose prime exponents are odd and different from $9$. Clearly, any $X$ in the conjecture is a subset of $Y$. Yet, the density of $Y$ equals

$$ \frac{1}{2}\prod_{p>2}\left(1-\frac{1}{p}\right)\left(1+\frac{1}{p-p^{-1}}-\frac{1}{p^9}\right) $$

which is less than

$$ 0.42264954363092750400907132916 $$

by the SAGE command

```
(1/2)*prod([RealField(100)(1-1/p)*(1+1/(p-1/p)-1/p^9) for p in prime_range(3,1000000)])
```