Is it possible to unite two $n$-vertex trees such that the resulting graph has bounded tree-width?

Formally, does there exists a constant $k$ such that given two $n$-vertex trees $T_1$ and $T_2$ there exist labeled trees $T_1'=([n], E_1)$ and $T_2' =([n],E_2)$ with $T_1' \simeq T_1$, $T_2' \simeq T_2$ and $$ \texttt{treewidth} (T_1' \cup T_2') \leq k, $$ where $T_1' \cup T_2' = ([n], E_1 \cup E_2)$.

**Special cases**

- $T_1 \simeq P_n$ and $T_2 \simeq P_n$. While one can 'construct' a grid out of two large paths (and a grid has 'large' tree-width), $T_1$ and $T_2$ can be united into a path $P_n$ (which has treewidth 1).
- $T_1$ is a star, $T_2$ is an arbitrary tree. In this case, no matter how we unite the trees we get a chordal graph with maximum clique size 3, and therefore of tree-width 2.
- It is also possible to unite a path with an arbitrary tree into a graph of bounded tree-width. But it is not clear if this is always possible for a pair of arbitrary trees.

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