Baudon's neighbor full sum distinguishing total coloring conjecture

Let $G$ be a nice graph and let $w:V(G)\cup E(G)\rightarrow\{1,2,\ldots,k\}$ be a total integer weighting. For each vertex $v$, define

$$\sigma^{ven}(v):=w(v)+\sum_{e\ni v}w(e)+\sum_{u\in N(v)}w(u),$$

where $N(v)$ is the open neighborhood of $v$. The weighting is a neighbor full sum distinguishing total coloring when adjacent vertices receive distinct values of $\sigma^{ven}$, and $\chi^{ven}(G)$ is the least such $k$. Baudon's conjecture. For every nice graph $G$, $\chi^{ven}(G)\leq 3$. This extends total neighbor sum distinguishing colorings by also including the weights of neighboring vertices. The source presents the assertion as open.