  Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total
$k$-coloring of $G$. Define a weight function on total coloring as
$$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where
$N(x)=\{y\in V(G)|xy\in E(G)\}$. If $\phi(x)\neq \phi(y)$ for any edge $xy\in
E(G)$, then $f$ is called a neighbor full sum distinguishing total $k$-coloring
of $G$. The smallest value $k$ for which $G$ has such a coloring is called the
neighbor full sum distinguishing total chromatic number of $G$ and denoted by
fgndi$_{\sum}(G)$. The coloring is an extension of neighbor sum distinguishing
non-proper total coloring. In this paper we conjecture that
fgndi$_{\sum}(G)\leq 3$ for any connected graph $G$ of order at least three. We
prove that the conjecture is true for (i) paths and cycles; (ii) 3-regular
graphs and (iii) stars, complete graphs, trees, hypercubes, bipartite graphs
and complete $r$-partite graphs. In particular, complete graphs can achieve the
upper bound for the above conjecture.
