Suppose you have a propositional formula $\varphi$ in CNF. You want to efficiently obtain an equisatisfiable CNF formula encoding $\neg \varphi$. You use the usual Tseitin encoding with auxiliary variables. Now,
Question. If the primal treewidth of $\varphi$ is $k$, what is the primal treewidth of $\neg\varphi$ after transforming into CNF via the Tseitin procedure? Is it bounded by some function of $k$, or does it depend on the entire size of $\varphi$?
Here by primal treewidth I mean the treewidth of the primal graph of $\varphi$, that is, the graph where the nodes are the variables of the formula and two nodes are connected if the corresponding variables appear together in some clause.