The nearly connected partition conjecture

Let $G$ be a 2-connected graph of order $n$, and let $n=n_1+n_2+\cdots+n_k$ be a partition of $n$. A subset $A\subseteq V(G)$ is nearly connected if it is contained in a subtree $T$ of $G$ of order at most $|A|+1$. The nearly connected partition conjecture. The vertex set $V(G)$ can be partitioned into nearly connected parts $A_1,\ldots,A_k$ such that $|A_i|=n_i$ for $1\leq i\leq k$. The paper proves the equal-size case with all parts of size $4$, while the assertion for arbitrary specified part sizes remains open.