For real numbers $x$, let $\|x\|$ denote the distance from $x$ to the nearest integer. Define a function $\ell:\mathbb{R}^3\rightarrow\mathbb{R}$ by $$\ell(\alpha,\beta,\gamma)=\liminf_{n\rightarrow\infty}n\|n\alpha\|\|n\beta\|\|n\gamma\|.$$ Does $\ell(\sqrt2,\sqrt3,\sqrt6)=0$?
Edit:
I went looking for small values of $\ell(\sqrt2,\sqrt3,\sqrt6)$ and could not find a $n$ resulting in a quantity smaller than .0000000263.
Plotting two of the three small scaled errors I computed during this hunt for small values did give the following rather nice fingerprint of how the 150,000 or so errors I found are distributed:
So how does this image relate to the original question? Well, $\ell(\sqrt2,\sqrt3,\sqrt6)=0$ exactly when the graph of the scaled errors has a limit point on some axis. Looking at the pic above, which is not the usual and expected furball of points, this conjecture at least looks plausible.