Limit periodicity conjecture for Jacobi matrices from expanding polynomials

Let $T(z)$) be an expanding polynomial of degree $d$ with a real Julia set $E$, where $E\subset[-\xi,\xi]$ and $T^{-1}:[-\xi,\xi]\to[-\xi,\xi]$. For $x\in[-\xi,\xi]$, define the Jacobi matrix $J=J(x)$ by

$$\left\langle 0\left|(z-J(x))^{-1}\right|0\right\rangle=\frac{T'(z)/d}{T(z)-x}.$$

Let $J_n(x)$ be the Jacobi matrix associated with the iterate $T_n=T^{\circ n}$, whose degree is $d_n$. Limit periodicity conjecture. For every $\epsilon>0$, there exists $n$ such that

$$\left\|J_n(x)-J_n(0)\right\|\leq\epsilon.$$

This conjecture concerns the asymptotic independence of the Jacobi matrices from the parameter $x$ under iteration of an expanding polynomial with real Julia set. The surrounding discussion notes that the analogous limit-periodicity problem is open even for quadratic polynomials with $C>2$, so the resolution status of this formulation is not established in the supplied text.