The alternative-measure conjecture for the polynomial $1+x_1+3+x_k$

Let $c(P)$ denote the alternative measure of a polynomial, with the last variable used as the subscript, and let $x_1,3,x_k$ be variables. Numerically, for $k\leq 6$, the alternative measure is computed from

$$c_k(1+x_1+x_2+\cdots+x_k):=\int_0^1\cdots\int_0^1\frac{e^{2\pi i t_k}}{1+e^{2\pi i t_1}+e^{2\pi i t_2}+\cdots+e^{2\pi i t_k}}\,dt_1\cdots dt_k.$$

Alternative-measure conjecture.

$$c(1+x_1+x_2+\cdots+x_k)=\frac{1}{k+1}.$$

The calculation for three variables gives $c(1+x+y+z)=1/4$, and the numerical evidence suggests this formula for the multivariate family.