Let $p$ be a prime and let $F$ be a free group of rank $d\geq 1$. Kostrikin [1] proved that the $d$-generated Burnside group $B=B(d,p)=F/F^p$ of exponent $p$ has a maximal finite quotient $\overline{B}=\overline{B(d,p)}=B/N$ where $N$ is the intersection of all the subgroups of $B$ of finite index. I understand that Kostrikin proved that the nilpotency class $c(d,p)$ of $\overline{B(d,p)}$ has an upper bound.

Question 1: Are lower bounds known for $c(d,p)$?

Question 2: Is $c(d,p)\geq d$ for $d\leq 4$ and $p>d$?

Question 3: For which $n$ is the $n$th section of the lower exponent-$p$ central series for $\overline{B(d,p)}$ isomorphic to the $n$th Lie power $L^n(V)$ where $V=({\mathbb F}_p)^d$?

[1] Kostrikin, A. L: On Lie rings with Engel's condition. Dokl. Akad. Nauk SSSR 108, no. 4 (1956) 580-582.