Is the following statement true?

Let $m,n,d$ be natural numbers. Then there exists a natural number $D=D(d,m,n)$ with the following property: If a polynomial $P(x_1,\dots,x_n)$ of total degree $d$ could be represented as $P=g(h_1,\dots,h_m)$ where $g=g(y_1,\dots,y_m)$ and $h_i=h_i(x_1,\dots,x_n)$ are polynomials, then there exists another representation $P=\tilde{g}(\tilde{h}_1,\dots,\tilde{h}_m)$ in which the total degree of any of the constituent parts of the RHS is at most $D$.

It is possible to come up with trivial examples such as $g\equiv 0$ or $h_1,\dots,h_m\equiv 0$, or $h_1= h_2$ and $g(y_1,y_2)=y_1-y_2$ where the degree of $g(h_1,\dots,h_m)$ is small but $\max\{\deg g,\deg h_1,\dots,\deg h_m\}$ could be arbitrarily large. For this reason I am asking for an alternative decomposition formed by polynomials of bounded degrees.