The iterated-limit conjecture for alternative measures of sparse polynomials

Let $c(P)$ be the alternative measure of a polynomial, and let $k\geq 2$. Consider the integer-exponent family $1+x+x^{n_2}+x^{n_3}+\cdots+x^{n_k}$ and take the iterated limit with $n_2,n_3,\ldots,n_k$ tending to infinity in the displayed order. Iterated-limit conjecture.

$$\lim_{n_2\to\infty}\lim_{n_3\to\infty}\cdots\lim_{n_k\to\infty}c(1+x+x^{n_2}+x^{n_3}+\cdots+x^{n_k})=\frac{k-1}{k+1}.$$

The paper derives this as a conjectural consequence of the preceding alternative-measure formula; it generalizes the displayed three-variable value $1/2$.