Degeneracy conjecture for proper conflict-free degree-plus list coloring

A graph $G$ is $d$-degenerate if every subgraph of $G$ has a vertex of degree at most $d$. For a graph $G$, a $({\rm degree}+d)$-list assignment assigns each vertex $v$ a list $L(v)$ with $|L(v)|\geq d_G(v)+d$. A graph is proper conflict-free $({\rm degree}+d)$-choosable if it has a proper conflict-free $L$-coloring for every such list assignment. Degeneracy conjecture. If a connected graph $G\ne C_5$ is $d$-degenerate for some positive integer $d$, then $G$ is proper conflict-free $({\rm degree}+d)$-choosable. The paper proves the corresponding bound with $d+1$ in place of $d$; reducing it by one is presented as challenging and remains open.