The solution by Ilya Zakharevich:
... \\ \cline{2-4}
\noalign{\vskip\doublerulesep
\vskip-\arrayrulewidth} \cline{2-4}
...
results in this for me:
Note that I have a vertical line on the very left which I would want to stay contiguous, so this is not so nice.
So my solution:
... \\ \cline{2-4}
\noalign{\vskip-2\tabcolsep \vskip-3\arrayrulewidth \vskip\doublerulesep}
\\ \cline{2-4}
...
This adds another row (\\
) and then tries to reduce the space of the row (I probably messed up the calculation...).
I came up with another variant:
... \\ \cline{2-4}
\multicolumn{4}{|c}{\phantom{x}}
\vspace{-2\tabcolsep}\vspace{-3\arrayrulewidth}\vspace{\doublerulesep}
\\ \cline{2-4}
...
Which results in: