In my MSE question, "Conjectured connection between $e$ and $\pi$ in a semidisk", the answer included

$$\prod_{k=1}^\infty\left[1-\big((k+1)^{1/3}-k^{1/3}\big)^3\right]\approx 0.96454\ldots.$$

Does this infinite product have a closed form?

By "closed form", I mean an expression in terms of known constants. I realize that this is still subjective, but I have found that certain kinds of geometrical constructions yield infinite products with closed forms (example1, example2, example3). The infinite product in this question also comes from one of these kinds of geometrical constructions, so I wonder if it also has a closed form.