Exchange-related braids have equal first-order Q-polynomials

Let $\beta$ be a braid, and let ${\bf Q}_{\widehat\beta,1}(x)$ denote the first-order coefficient of the ${\bf Q}$-polynomial of its closure $\widehat\beta$. Two braids are exchange related when they are related by an exchange move.

Exchange-vanishing conjecture. The polynomials ${\bf Q}_{\widehat\beta,1}(x)$ of exchange-related braids vanish in the same way as the Fiedler polynomial; in particular, the first-order ${\bf Q}$-polynomial does not distinguish exchange-related braids.

The preceding computation shows equality of the first-order terms for the displayed exchange-related pair, motivating the proposed generalization of the corresponding property of the Fiedler polynomial. The source does not provide a resolution of the claim.