Rado conditions conjecture for nonlinear Diophantine equations with linear diagonal polynomial

Let $P\in \mathbb{Z}[x_{1},\ldots,x_{n}]$ be a polynomial, and define its corresponding monovariate polynomial by

$$\widetilde{P}(w):=P(w,\ldots,w).$$

Assume that $\widetilde{P}$ is a nonzero homogeneous linear polynomial and that the equation $P(x_{1},\ldots,x_{n})=0$ has infinitely many solutions in $\mathbb{N}$. The Rado conditions conjecture. The polynomial $P$ is partition regular if and only if it satisfies both the maximal Rado condition and the minimal Rado condition. These conditions are known to be necessary in the stated setting; the conjecture asserts their sufficiency, extending Rado's characterization of partition-regular homogeneous linear equations to this class of nonlinear Diophantine equations.