Characterization of simulation rates for generalized Lipschitz functions

Let $\alpha\in\mathbb{N}$, and let $f\in {C^\alpha}^*[0,1]$ be a function bounded strictly between $0$ and $1$. Here ${C^\alpha}^*[0,1]$ consists of functions that are $r:=\alpha-1$ times differentiable and whose $r$th derivative has symmetric modulus of continuity

$$\omega^*(f^{(r)},h):=\sup_{t<h,\;x\in[t,1-t]}\left|f^{(r)}(x+t)-2f^{(r)}(x)+f^{(r)}(x-t)\right|$$

of order $O(h)$. Write $\Delta_n(x)$ for the approximation scale used in the paper. Simulation-rate conjecture. The function $f$ can be simulated at the rate $(\Delta_n(x))^\alpha$ on $[0,1]$: there exist polynomials $g_n$ and $f_n$ satisfying conditions (i)--(iv) of the paper's reduction result and the bound specified there as (app_order). This conjecture proposes the analogue, for simulation rates, of the known characterization of unrestricted polynomial approximation by the class ${C^\alpha}^*[0,1]$.