The odd-degree peculiar linear mean relationship

Let $f$ be an odd-degree polynomial of degree $D$, and let $\varphi_{D,0,\rho}$ denote the quantities used in the source for indices $D$, $0$, and $\rho$. Odd-degree peculiar linear mean relationship. For all odd-degree polynomials of degree $D$, one has

$$\sum_{0<\rho<D}(-1)^\rho\binom{D}{\rho}\,\varphi_{D,0,\rho}=0.$$

This is presented as a main computational result, supported by computations through degree $49$. The surrounding discussion reports a two-dimensional family of fundamental linear relationships in odd degrees beyond $3$, but does not provide a proof of this particular formula.