  Let $\mu_n$ be the standard Gaussian measure on $\mathbb{R}^n$ and $X$ be a
random vector on $\mathbb{R}^n$ with the law $\mu_n$. U-conjecture states that
if $f$ and $g$ are two polynomials on $\mathbb{R}^n$ such that $f(X)$ and
$g(X)$ are independent, then there exist an orthogonal transformation $L$ on
$\mathbb{R}^n$ and an integer $k$ such that $f\circ L$ is a function of
$(x_1,\cdots,x_k)$ and $g\circ L$ is a function of $(x_{k+1},\cdots,x_n)$. In
this case, $f$ and $g$ are said to be unlinked. In this note, we prove that two
symmetric, quasi-convex polynomials $f$ and $g$ are unlinked if $f(X)$ and
$g(X)$ are independent.
