I asked this question on math.se and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero.

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 combination of monomials of the form $x_{i_1}\cdots x_{i_d}$ with $0\le i_1<i_2<\cdots<i_d\le n$.

Given an ideal $I=(f_1,\ldots,f_r)\subseteq\Bbbk[x_0,\ldots,x_n]$ with the property that the $f_i$ are irreducible, homogeneous, multilinear polynomials of (pairwise) different degrees, I am asking whether $I$ is radical.

I actually don't believe it holds in general - if this is the case, I would love to see a counterexample.

If it is true however, then I am sure that the assumption on the degree can not be dropped (see this example of an ideal generated by irreducible, homogeneous, multilinear polynomials which is not radical). I would also love to see a proof in this case, of course.

Thanks a lot in advance!