Conjecture on Smarandache cube-digital subsequences

Let the Smarandache cube-digital subsequence consist of the cubes whose decimal digits form a subsequence of the digits of the cube itself. The conjecture concerns all terms of this subsequence and the parameters $D$ and $k$ in the following form.

Smarandache cube-digital subsequence conjecture. All terms of the Smarandache cube-digital subsequence are of the form $D \times 10^{3k}$ where $D \in \left\{0,1,8\right\}$ and $k \in \mathbb{N}_0$.

Only the trivial terms had been found up to the computational bound $10000^3$; the assertion remains unproved in the supplied text.