The representation theorem for odd or even involutive FL \(_e\) -chains via bunches of layer groups, as presented in [12], has been refined to reveal a more direct constructional relationship between these chains and bunches of layer groups, entirely circumventing the intermediary notion of layer algebras. Leveraging this refinement, it is demonstrated that both the variety of semilinear odd involutive FL \(_e\) -algebras and its idempotent symmetric subvariety admit densification. Finally, by applying the algebraic techniques developed in [14], the strong standard completeness of Involutive Uninorm Logic with Fixed Point ( \(\textbf{IUL}^{fp}\) ) is established, thereby strenghtening the main result of [11].
