Logarithmic improvement conjecture for defective colouring of linear hypergraphs

Let $r\geqslant 1$, let $\Delta$ be the maximum degree, and let $d\geqslant 2$. A hypergraph is linear if every two distinct edges intersect in at most one vertex. A $d$-defective $k$-colouring is a colouring of the vertices with $k$ colours such that every colour class induces maximum degree at most $d$.

Defective-colouring conjecture. Every $(r+1)$-uniform linear hypergraph with maximum degree at most $\Delta$ has a $d$-defective colouring with

$$\nk=\mathcal{O}\left(\left(\frac{\Delta}{\log\Delta}\cdot\frac{\log d}{d}\right)^{1/r}\right).$$

This would match the lower bound obtained from known examples of linear hypergraphs, up to the implicit constant, and would establish a logarithmic improvement over the general defective-colouring bound for sparse hypergraphs. The conjecture is presented as open in the source.