Classification of linear period 1 seed polynomials with positive coefficients

Let $n$ be a positive integer, and let $P$ be a linear polynomial in $x_1,\ldots,x_n$ with positive coefficients. Say that $P$ generates a period 1 seed when it generates a seed of period 1 in the Laurent phenomenon framework. Linear period 1 seed conjecture. If $n$ is odd, no linear polynomial with positive coefficients generates a period 1 seed. If $n$ is even, the only linear polynomial $P$ with positive coefficients that generates a period 1 seed is

$$P=x_{n/2}+1.$$

This conjecture proposes a complete classification of linear polynomials with positive coefficients that generate period 1 seeds, distinguishing the odd- and even-dimensional cases. The supplied source does not state a resolution, so the conjecture remains open.