Claims \[clm:0\] and \[clm:3\], $H(x)$ has at least $k\ell-1$ leaf-blocks that have pairwise distinct pre-leaves. Therefore Claim \[clm:1\] and \[clm:2\] yield at least $k\ell-1$ distinct vertices $y \in X$ such that $H(x)$ is joined to $H(y)$ by an edge of $G$. Now let $G^*$ denote the graph obtained from $G$ by contracting $H(x)$, for each $x\in X$. It follows that the minimum degree of $G^*$ is at least $k\ell-1$ and hence, by Lemma \[lem:disjointlongcycles\], $G^*$ contains $k$ vertex-disjoint long cycles. This in turn implies that $G$ contains $k$ edge-disjoint long cycles; contradiction.\
It remains to prove Claims \[clm:0\], \[clm:1\], \[clm:2\] and \[clm:3\]. For that, the following two auxiliary claims are useful.

\[clm:noleafblock\] Let $x \in X$. No block of $H(x)$ is a leaf-block of $G$.

Suppose for a contradiction that some block $L$ of $H(x)$ is a leaf-block of $G$. Then every cycle in $G$ that visits an edge of $L$ must in fact be a subgraph of $L$. However, $L$ is a subgraph of $H(x)$ and thus contains no long cycle. We conclude that every long cycle of $G$ must be edge-disjoint from $L$. Hence the multigraph obtained from $G$ by contracting $L$ to a vertex must also be 