The global velocity-properties conjecture for the massless XXZ chain

Let $\zeta\in(0,\pi)$ and $p_F\in[0,\pi/2]$. Let $\mathfrak{v}_1$ be the velocity of $1$-string particles and holes, and let $\mathfrak{v}_r$ be the velocity of an $r$-string excitation. Proposition 1 asserts, among other properties, the existence and smoothness of the velocity-matching maps $\mathfrak{t}$, $\mathfrak{p}_L$, $\mathfrak{p}_R$, and $\mathfrak{h}^{(r)}$, together with the stated monotonicity, uniqueness, and velocity-range conclusions.

Global velocity-properties conjecture. The conclusions of Proposition 1 hold for all

$$\zeta\in(0,\pi),\qquad p_F\in[0,\pi/2].$$

These properties are supported by numerical analysis for the displayed parameter ranges; the source does not provide a proof, so their general validity remains open.