Upper bound for Hall's conjecture on separation of squares and cubes

Hall's (weak) conjecture is the statement that for all $$\varepsilon > 0$$ there exists a positive number $$c(\varepsilon) > 0$$ such that for all $$x,y \in \mathbb{Z}$$ with $$y^2 \ne x^3$$, that

$$\displaystyle \left \lvert y^2 - x^3 \right \rvert \geq c(\varepsilon) |x|^{\frac{1}{2} - \varepsilon}.$$

My question concerns a related question. What is the smallest $$\theta \geq 0$$ for which it is known that there exist infinitely many $$x,y \in \mathbb{Z}$$ with $$y^2 \ne x^3$$ satisfying

$$\displaystyle |y^2 - x^3| \ll |x|^{\frac{1}{2} + \theta}?$$

The choice $$\theta = 1/2$$ can be achieved as follows: write $$x^3 = (x-1)^2(x+2) + 3x - 2$$, and choose $$x = t^2 - 2$$ and $$y = (t^2 - 3)t$$. Then

$$y^2 - x^3 = (t^2 - 3)^2t^2 - (t^2 - 3)^2 t^2 + 3(t^2 - 2) - 2 = 3t^2 - 8,$$

which is manifestly $$O(t^2) = O(|x|)$$. Here we can freely choose $$t \in \mathbb{Z}$$, so this gives infinitely many integers satisfying the required inequality.

Can one do better than $$\theta = 1/2$$?

The best $$\theta$$ is $$0$$. It is known that there are infinitely many solutions of 0 < $$|x^3 - y^2| \ll x^{1/2}$$, parametrized by certain "Pell equations"; indeed one such family attains $$|x^3 - y^2| \sim C x^{1/2}$$ with $$C = 5^{-5/2} 54 \approx 0.966$$. See [D]. The underlying polynomial identity $$(t^2 + 10t + 5)^3 - (t^2 + 22t + 125)(t^2 + 4t − 1)^2 = 1728t$$ is connected with the degree-6 cover $${\rm X}_0(5) \to {\rm X}(1)$$ of modular curves [E].
[D] Danilov, L.V.: The Diophantine equation $$x^3 - y^2 = k$$ and Hall's conjecture, Math. Notes Acad. Sci. USSR 32 (1982), 617-618.
[E] Elkies, N.D.: Rational points near curves and small nonzero $$|x^3-y^2|$$ via lattice reduction, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 (arXiv: math.NT/0005139).