A four-term representation by pentagonal quadratic polynomials

Let $\mathbb N$ denote the nonnegative integers. Four-term pentagonal representation conjecture. Every integer $n>51$ can be written as

$$n=\frac{w(5w-1)}2+\frac{x(5x-1)}2+\frac{y(5y+1)}2+\frac{z(5z+1)}2$$

with $w,x,y,z\in\mathbb N$.

This is an explicit eventual universality claim analogous to Lagrange’s four-square theorem, with nonnegative variables and two signs in the linear terms. No resolution is stated in the source.