Containment of the (k+1)-core in the giant k-deeply connected component

Determine whether the (k+1)-core of the random graph lies entirely within the giant k-deeply connected component with high probability.

Background

The paper introduces k-deeply connected components as maximal induced subgraphs containing k edge-disjoint spanning trees. For c above the emergence threshold, it proves the existence and asymptotic size of a unique giant k-deeply connected component in the sparse random graph.

The paper also relates the asymptotic size of this giant component to the asymptotic size of the (k+1)-core, but does not establish whether the core is entirely contained in the giant component. The authors explicitly identify this containment question as unresolved.