Can experts roll dice with non-uniform distribution?
In theory, yes (the macro nature of the dice and the table overcoming any quantum-level randomness, leaving you with classical physics). In practice, the bulk of the evidence says no, that the chaos in the system is greater than human skill can overcome, but some people claim such a special skill.
See the short wiki article for (the scant) details.
In response to the addition to your question: &quot;is there statistical proof&quot;? The answer is (in 2018, at least): &quot;no&quot;, according to this article in the UNLV Gaming Research &amp; Review Journal

We do not assert whether dice control is possible or not (there is a lack of published evidence).

This assumes that you throw them in a standard way. If you &quot;throw&quot; a die in an obviously forced way (pick it up and drop it from a small height with no shake and no bounce), then obviously the side that was on top in your hand has a greater likelihood of staying on top. Any throw of a die with the usual features of a throw (shake and roll, true roll) will be too chaotic to control.
Exception: for n=2 in the particular expression of a coin, it is possible to &quot;not flip&quot; the coin but instead make it wobble in the air, fooling most people and yet resulting in significant deviation from fair. Again, this deviates from a true &quot;roll&quot; of the die, but unlike other faux rolls, this one can be mistaken for a true roll.