Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of degree $r$ each of whose terms involves at most $d−1$ variables?

Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of degree $r$ each of whose terms involves at most $d−1$ variables?

The linear polynomials are truly linear functions: e.g. $p(x) = w^T x,$ so have no constant terms. And the field is $\mathbb{R}.$

As an example, this is true for degree two polynomials, because you can show that given an arbitrary $A$, there exists a nonnegative matrix $B$ and a diagonal matrix $D$ such that the quadratic forms corresponding to $A$ and $B+D$ are identical.