The tree conjecture for maximum running time

For a tree $T$ on $t$ vertices, define

$$M^*(t):=\max\left\{\limsup_{n\rightarrow\infty}M_T(n):T\text{ is a $t$-vertex tree}\right\}.$$

Tree conjecture. For all $t\in\mathbb N$,

$$M^*(t)=t-1.$$

The known upper bound is quadratic in $t$, while stars give the lower bound $M^*(t)\geq t-1$. The conjecture asserts that stars attain the largest asymptotic maximum running time among $t$-vertex trees.