Preservation of the SSP under attaching a leaf

Determine whether, for every SSP graph G with a leaf v, adjoining a new vertex w by the single edge {v,w} always produces an SSP graph H.

Background

The paper observes that the SSP of a graph is not preserved by arbitrary vertex addition: K_{1,3} is an SSP graph whereas K_{1,4} is not. In contrast, the authors’ results for trees, girth-three unicyclic graphs, and the tadpole families T_{4,n} and T_{5,n} indicate that extending a pending path can preserve the SSP in several settings. The question asks whether this preservation holds universally when a new leaf is attached to an existing leaf.

References

Is it true that for any $G\in $ having a leaf $v\in V(G)$, every graph $H$, obtained as $$V(H)=V(G)\cup \{w\} \text{ and } E(H)=E(G) \cup \{\{v,w\}\}$$

is also in the set $$?

— The strong spectral property for some families of unicyclic graphs  (2501.01719 - Koljančić et al., 3 Jan 2025) in Question labeled q:leaf-to-leaf, Section 6, Further examples and questions