Invariance conjecture for the proposed expression of the expectation value

Let $Y_L$ be the quantity defined in the paper from the finite-size expectation-value calculation, and let $q$ be the model parameter. Let $w$ be its spectral variable. Invariance conjecture. The proposed expression for $Y_L$ is invariant under

$$w\longrightarrow q/w.$$

This invariance is used to establish the relations between $Y_L$ and the marked quantities $X_L^{(k)}$ and to complete the degree arguments. The supplied text does not state a proof or a resolution beyond identifying the invariance as a property of the proposed expression.