Are all (graded) Artinian complete intersections like this?

I'm trying to prove some stuff (it's not important what) about (graded) Artinian complete intersections $R=\mathbb{C}[x_1,\ldots,x_n]/I$, where the $x_i$ have certain positive weights and where $I$ is generated by a regular sequence of weighted homogeneous polynomials $f_1,\ldots,f_n$, and I find I'd like the following statement to be true (up to isomorphism of $R$, of course):

Claim One may choose generators $f_i$ for $I$ such that each $f_i$ is of the form $x_j^k+g(x_1,\ldots,x_n)$, where the $x_j$-degree of $g$ is less than $k$.

Empirically this seems to be true (every example of an Artinian c.i. I know is of this form), but I have no idea how to prove it. It does somehow seem fishy to demand that the $f_i$ each have weighted degree a multiple of the weighted degree of a single generator $x_j$, but I can not for the life of me find a counterexample.

Is my Claim known to be true/false? How would I go about trying to prove it? In the case that the $x_i$ have equal weights it's not too hard to get this by linear change of variables, but in the general case I don't see how to proceed.