A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex of $G$. The least numbers of colours in such colourings of $G$ are denoted $χ_{\rm pcf}(G)$ and $χ_{\rm pcf}^h(G)$, respectively. It is known that $χ_{\rm pcf}^h(G)$ can be as large as $(h+1)(Δ+1)\approx Δ^2$ for graphs with maximum degree $Δ$ and $h$ very close to $Δ$. We provide several new upper bounds for these parameters for graphs with minimum degrees $δ$ large enough and $h$ detached from $δ$. In particular we show that $χ_{\rm pcf}^h(G)\leq (1+o(1))Δ$ if $δ\gg\lnΔ$ and $h\ll δ$, and that $χ_{\rm pcf}(G)\leq Δ+O(\ln Δ)$ for regular graphs. These specifically refer to the conjecture of Caro, Petruševski and Škrekovski that $χ_{\rm pcf}(G)\leq Δ+1$ for every connected graph $G$ of maximum degree $Δ\geq 3$, towards which they proved that $χ_{\rm pcf}(G)\leq \left\lfloor\frac{5Δ}{2}\right\rfloor$ if $Δ\geq 1$.