Is an ideal generated by multilinear polynomials of different degrees always radical?

Is an ideal generated by multilinear polynomials of different degrees always radical? References Definition. A polynomial $f\\in\\Bbbk[x_0,\\ldots,x_n]$ is called multilinear if $\\deg_{x_i}(f)=1$ for each $0\\le i \\le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of degree $d$, then $f$ is a linear combintation of monomials of the form $x_{i_1}\\cdots x_{i_d}$ with $0\\le i_1