Asymptotic decomposition conjecture for the XX-chain time-like correlation ratio

Let $r(m,t,h):=(-1)^m\langle \sigma^-_{1}(0)\sigma^+_{m+1}(t)\rangle_{-h,T}/I_m(t)$ be the ratio formed from the time-like correlation function and its leading asymptotic factor. For fixed $T$, $h$, and $m$, consider $t\gg 1$. Asymptotic decomposition conjecture. The ratio consists of non-oscillating and oscillating parts,

$$r(m,t,h)\sim r_{\rm non-osc}(m,t,h)+r_{\rm osc}(m,t,h).$$

The non-oscillating part includes $A_0(T,h)$. If $2\pi/\omega$ is the period of the oscillating part, then

$$\omega\sim h_c+h\quad\text{as }h\rightarrow-h_c,$$

and its amplitude is expanded in powers of $t^{-1/2}$. This is a numerical proposal based on calculations with $n=1536$; the asymptotic correction to $A_0(T,h)$ and the oscillating amplitude are described only at the stated power-counting level.