Asymptotic expansion conjecture for Takeuchi numbers

Let $T_n$ denote the Takeuchi numbers, let $W$ be the Lambert $W$-function, and set $w=W(n)$. Let $C_T$ be a positive real constant. Asymptotic expansion conjecture. As $n$ tends to infinity, one has

$$\log T_n=e^w(w^2-w+1)+\frac{1}{2}w^2-\frac{1}{2}\log(1+w)+\log C_T-1-\frac{w(26w^2+67w+46)}{24(1+w)^3}e^{-w}+O(e^{-2w}).$$

This conjecture gives an explicit asymptotic expression for the Takeuchi numbers in terms of $w=W(n)$ and a positive constant $C_T$; the preceding numerical evidence suggests the stated form and estimates $C_T\approx2.239433104005260731754785$.