Upper-bound conjecture for normal form limit values

For every odd integer $t\geqslant 9$, let $N_t$ be the index

$$N_t=t+4L_t+3,$$

where

$$L_t=\max\left\{\ell\in\mathbb{N}_0:t>u_\ell\right\}=\left\lceil\frac{-t-2+\sqrt{5t^2-4t-12}}{4}\right\rceil-1,$$

and let $m_t$ be the limit of the normal form orbit and $y_{N_t}$ its term at index $N_t$. The upper-bound conjecture.

$$m_t<y_{N_t}.$$

The paper has proved lower bounds for $m_t$ and the transit time and reports that available evidence suggests an upper bound by the largest term in the regular phase. The stated inequality remains conjectural here.