The exact growth conjecture for treedepth in $P_t$-free graphs

For integers $t\geqslant 2$ and $k\geqslant 2$, define

$$g(k,t)\coloneqq\max\{\operatorname{td}(G):\operatorname{tdtwo}(G)\leqslant k\text{ and }G\text{ is }P_t\text{-free}\}.$$

Here, $\operatorname{td}(G)$ is the treedepth of $G$, $\operatorname{tdtwo}(G)$ is its 2-treedepth, and $P_t$ denotes the path on $t$ vertices. Exact growth conjecture. For all integers $t\geqslant 2$ and $k\geqslant 2$,

$$g(k,t)=\binom{\lfloor (t-1)\slash 2\rfloor+k-1}{\lfloor (t-1)\slash 2\rfloor}.$$

The paper establishes that, for fixed $t$, $g(k,t)=\Theta(k^{\lfloor(t-1)\slash 2\rfloor})$ and determines $g(k,4)$ and $g(k,5)$ exactly. The displayed formula is proposed as the general exact value and remains open beyond the cases settled in the paper.