This update of question asked before.

Let $n$ be a natural number. Consider a subvariety in $\mathbb A^{3n+2}$ (say over $\mathbb C$) given by the equation

$$x_1(t-y_1)\dots (t-y_n)+x_2(t-z_1)\dots(t-z_n)-(t-w_1)\dots(t-w_n)=0\in \mathbb C[t].$$

Here $x_i,y_i,z_i,t_i$ are variables and equation means equality of polynomials in $t$. So we have $n+1$ equations. (For instance $x_1+x_2=1$.) If we divide by $(t-w_1)\dots(t-w_n)$ we get a $S$-unit equation. This $S$-unit equation has close relations for functional equations for di- and tri-logarithms.

Is this variety unirational?

In the case $n=2$ one can write down equations explicitly. We get a singular cubic hypersurface which is not a cone. So it is rational by well-known result. But what happens for arbitrary $n$?

Any comments are welcome.