The max-weight connected subgraph problem (MWCS) may be described as follows: given a simple graph $G=(V,E)$ and a weight function $w:V\to\mathbb{R}$, one seeks for a subset $L\subseteq V$ for which the spanned subgraph $G[L]$ is connected and $\sum_{v\in L}w(v)$ is maximal.
The MWCS problem is known to be NP-hard (see https://arxiv.org/abs/1409.5308).
Consider the following variation: in the same settings, one seeks for a vertex set $L$ for which the spanned subgraphs $G[L]$ and $G[V\smallsetminus L]$ are both connected, and $\sum_{v\in L}w(v)$ is maximal.
My question is the following: is it possible that this variation takes the problem out from the NP class? If not - is it easy to see that it is still hard, given that the original problem is hard?
And does it make a difference if one asks $\sum_{v\in L}w(v)-\sum_{u\in V\smallsetminus L}w(u)$ to be maximal?