nt.number theory - A notion of independence for beatty sequences

Let $lfloorcdotrfloor$ denote the floor function. Is it true that for every positive integer $n$, there are real algebraic numbers $alpha_1ldotsalpha_n$ such that for every non-zero polynomial $pinmathbb{Z}[x_1ldots x_n]$, the equation
$$p(lfloor alpha_1 xrfloor ldots lfloor alpha_n xrfloor )=0$$
has only finitely many solutions in integers $x$?

The statement is true for $n=2$. In fact, no equation $p(x,lfloor 2^{1/3}x rfloor )=0$ can have infinitely many integer solutions. (Proof-Sketch: Suppose that, on the contrary, the equation had infinitely many solutions. Then $p(x,y)$ would have an irreducible factor $q(x,y)$ with infinitely many integer zeros, and the leading homogeneous part of $q(x,y)$ would be divisible by $x^3-2y^3$. But it is known that any irreducible polynomial $qinmathbb{Z}[x,y]$ with infinitely many integer zeros has leading homogeneous part a constant multiple of a power of a linear or quadratic form: See Theorem 21, Chapter 22 of Mordell's Diophantine Equations.)

Can anyone do $n=3$??