Neighbor sum distinguishing total choosability of planar graphs without intersecting 4-cycles

Given a simple graph $G$, a proper total-$k$-coloring $c:V\left(G\right)\cup E\left(G\right)$ $\to \{1,2,\dots ,k\}$ is called neighbor sum distinguishing if ${\sum }_c\left(u\right)\ne {\sum }_c\left(v\right)$ for any two adjacent vertices $u,v\in V\left(G\right)$, where ${\sum }_c\left(v\right)$ denote the sum of the color of $v$ and the colors of edges incident with $v$. The least number $k$ needed for such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by ${\chi }_{\Sigma }^{\prime \prime }\left(G\right)$. Pilśniak and Woźniak conjected that ${\chi }_{\Sigma }^{\prime \prime }\left(G\right)\le \Delta \left(G\right)+3$ for any simple graph $G$. Let $L_x(x\in V\left(G\right)\cup E\left(G\right))$ be a set of lists of real numbers and each of size $k$. The least number $k$ for which for any specified collection of such lists, there exists a neighbor sum distinguish total coloring of $G$ with colors from $L_x$ for each $x\in V\left(G\right)\cup E\left(G\right)$ is called the neighbor sum distinguishing total choosability of $G$, denoted by $c{h}_{\Sigma }^{\prime \prime }\left(G\right)$. In this paper, it is proved that $c{h}_{\Sigma }^{\prime \prime }\left(G\right)\le max\{\Delta \left(G\right)+3,10\}$ for any planar graph $G$ without intersecting 4-cycles.