Wedges of two non-ensnaring graphs

Prove that if two graphs are not ensnaring, then every graph obtained by wedging the two graphs at a single vertex is also not ensnaring.

Background

A wedge is formed by identifying one vertex of each of two graphs. The paper proves that wedging two ensnaring graphs always produces an ensnaring graph, but gives examples showing that the converse of this theorem is false when only one factor is ensnaring. The unresolved conjecture concerns the remaining case in which both factors are non-ensnaring.