branch of $S_{v_i}$, for $i \in k \setminus \{j\}$. In $V[G]$ let $g$ be the restriction of $f$ to the tree $(\bigotimes_{i \in k \setminus \{j\}} c_i) \bigotimes S_{v_j}$, which obviously is isomorphic to $S_{v_j}$. Then $q$ forces that $g$ is a club embedding from an isomorphic copy of $S_{v_j}$ into $U$, and $S_{v_j}$ is a Kurepa subtree of $T$. This contradicts (6).

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Data availability**

No data was used for the research described in the article.

**Acknowledgement**

The authors would like to thank the anonymous referee for careful comments and pointing out important mistakes in the previous version of the article.

The research on the second author is partially supported by grants from NSERC (455916), SFRS (7750027-SMART) and CNRS (UMR7586).

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