I shortened the probe pins of my experimental pipecap filter to 5mm in order to get rid of the unwanted response around 7GHz. As expected the filter is rather narrow now and the attenuation increases a lot.
Marcel made some new measurements up to 14GHz in order to see how the suppression behaves. It looks a lot better now but you can also see that at the upper end of the measurement range the attenuation is very low (keep in mind that the probes are nice quarter wavelength antennas there).
The following picture shows the filter tuned to the 6cm band:

pipecap 5mm probes 5760MHz

The passband attenuation is now always somewhere in the range of 2..3dB.

Tuned to the upper end of the possible range you see that it behaves more than a lowpass than a bandpass ;) The passband gets a bit wider.

pipecap 5mm probes 10610MHz

pipecap 5mm probes 11815MHz

I would assume that it makes most sense to design the probes beeing quarter lambda for the frequency were the notch of the filter appears (or slightly above). Since this depends on the frequency you want to tune the filter to you need to consider that before you make the filter.

Pipecap filter dimensions