Improvement of the clique-sum upper bound

Determine whether the factor of $3$ in the upper bound $3k+18$ for the strong isometric path complexity of clique-sums can be removed, specifically whether the bound can be improved to $k+c$ for some constant $c$.

Background

Theorem 3.2 proves that a clique-sum of finitely many graphs with strong isometric path complexity at most $k$ has strong isometric path complexity at most $3k+18$. The paper notes that clique-sums can strictly increase the parameter, so the unresolved issue concerns the optimal form of the general upper bound rather than preservation itself.