Odd-power symmetric sum conjecture

Let $m\in\mathbb{N}$ and let $x$ be a positive integer. The coefficients ${\mathbf{A}}_{m,0}, {\mathbf{A}}_{m,1},\ldots,{\mathbf{A}}_{m,m}$ are constants independent of $x$ and $k$. Odd-power symmetric sum conjecture. For every $m\in\mathbb{N}$, there exist coefficients ${\mathbf{A}}_{m,0}, {\mathbf{A}}_{m,1},\ldots,{\mathbf{A}}_{m,m}$ such that

$$x^{2m+1}=\sum_{k=1}^{x}\left({\mathbf{A}}_{m,0}k^0(x-k)^0+{\mathbf{A}}_{m,1}(x-k)^1+{\mathbf{A}}_{m,2}k^2(x-k)^2+\cdots+{\mathbf{A}}_{m,m}k^m(x-k)^m\right).$$

The claim proposes a generalization of the displayed decomposition of $x^3$ as a sum of symmetric terms involving $k$ and $x-k$. The source does not provide a resolution or further conditions on the coefficients, so its status remains open.