  Recently the first detections of highly ionised gas associated with two
Warm-Hot Intergalactic Medium (WHIM) filaments have been reported. The evidence
is based on X-ray absorption lines due to O VII and other ions observed by
Chandra towards the bright blazar Mrk 421. We investigate the robustness of
this detection by a re-analysis of the original Chandra LETGS spectra, the
analysis of a large set of XMM-Newton RGS spectra of Mrk 421, and additional
Chandra observations. We address the reliability of individual spectral
features belonging to the absorption components, and assess the significance of
the detection of these components. We also use Monte Carlo simulations of
spectra. We confirm the apparent strength of several features in the Chandra
spectra, but demonstrate that they are statistically not significant. This
decreased significance is due to the number of redshift trials that are made
and that are not taken into account in the original discovery paper. Therefore
these features must be attributed to statistical fluctuations. This is
confirmed by the RGS spectra, which have a higher signal to noise ratio than
the Chandra spectra, but do not show features at the same wavelengths. Finally,
we show that the possible association with a Ly-alpha absorption system also
lacks sufficient statistical evidence. We conclude that there is insufficient
observational proof for the existence of the two proposed WHIM filaments
towards Mrk 421, the brightest X-ray blazar on the sky. Therefore, the highly
ionised component of the WHIM still remains to be discovered.
