Zeeman's monotonicity conjecture for the Lyness rotation number

Consider the Lyness map with parameter $\alpha>0$ and let $\rho^v_\alpha$ denote its rotation number as a function of the level parameter $v$. Zeeman's conjecture. The function $\rho^v_\alpha$ is differentiable with respect to $v$ and satisfies: if $0<\alpha<1$, then $\rho^v_\alpha$ is strictly increasing in $v$; if $1<\alpha<\infty$, then $\rho^v_\alpha$ is strictly decreasing in $v$. This conjecture concerns the monotonicity of the rotation number for Lyness dynamics; the source attributes it to E. C. Zeeman (1996) and gives no evidence of a resolution.