The conjecture on infinitely many cubic-order approximations by rationals of the form $a/b^2$

Let $\alpha$ be a real number, and let $a,b$ be integers.

Conjecture on cubic-order $a/b^2$-approximations. There exists a constant $c(\alpha)$ such that

$$\left|\alpha-\frac{a}{b^2}\right|<\frac{c(\alpha)}{b^3}$$

has infinitely many integer solutions $(a,b)$.

This conjecture is presented as a classical conjecture supported by results for almost all real numbers and by numerical experiments. It concerns the expected sharp upper bound for approximation by rationals of the form $a/b^2$, between the known general estimates.