Regular-subhypergraph conjecture for linear hypergraphs

Let $k\geq 2$ be an integer, let $\lambda\geq 1$, and let a $k$-uniform linear hypergraph be a hypergraph in which every edge has $k$ vertices and any two distinct edges intersect in at most one vertex. Let $d$ be its average degree and let its maximum degree be at most $\lambda d$. Regular-subhypergraph conjecture. There exists a constant $c=c(k,\lambda)>0$ such that the hypergraph contains an $r$-regular subhypergraph for some

$$r\geq cd.$$

The conjecture would extend the paper's regularisation results from graphs to linear hypergraphs. The authors note that the needed hypergraph analogue of the Alon--Friedland--Kalai result is unknown, which is why the extension is conjectural.