  A graph $G=(V,E)$ is total weight $(k,k')$-choosable if the following holds:
For any list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of
$k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real
numbers, there is a proper $L$-total weighting, i.e., a map
  $\phi: V \cup E \to \mathbb{R}$ such that $\phi(z) \in L(z)$ for $z \in V
\cup E$, and $\sum_{e \in E(u)}\phi(e)+\phi(u) \ne \sum_{e \in
E(v)}\phi(e)+\phi(v)$ for every edge $\{u,v\}$. A graph is called nice if it
contains no isolated edges. As a strengthening of the famous 1-2-3 conjecture,
it was conjectured in [T. Wong and X. Zhu, Total weigt choosability of graphs,
J. Graph Th. 66 (2011),198-212] that every nice graph is total weight
$(1,3)$-choosable. The problem whether there is a constant $k$ such that every
nice graph is total weight $(1,k)$-choosable remained open for a decade and was
recently solved by Cao [L. Cao, Total weight choosability of graphs: Towards
the 1-2-3 conjecture, J. Combin. Th. B, 149(2021), 109-146], who proved that
every nice graph is total weight $(1, 17)$-choosable. This paper improves this
result and proves that every nice graph is total weight $(1, 5)$-choosable.
