The cosmological missing baryons at z < 1 most likely hide in the hot (T ≳ 10^5.5 K) phase of the warm hot intergalactic medium (WHIM). While the hot WHIM is hard to detect due to its high ionisation level, the warm (T ≲ 10^5.5 K) phase of the WHIM has been very robustly detected in the far-ultraviolet (FUV) band. We adopted the assumption that the hot and warm WHIM phases are co-located and therefore used the FUV-detected warm WHIM as a tracer for the cosmologically interesting hot WHIM. We performed an X-ray follow-up in the sight line of the blazar PKS 2155–304 at the redshifts where previous FUV measurements of O VI, Si IV, and broad Lyman-alpha (BLA) absorption have indicated the existence of the warm WHIM. We looked for the O VII Heα and O VIII Lyα absorption lines, the most likely hot WHIM tracers. Despite the very large exposure time (≈1 Ms), the Reflection Grating Spectrometer unit 1 (RGS1) on-board XMM-Newton data yielded no significant detection which corresponds to upper limits of log N(O VII(cm⁻²)) ≤ 14.5−15.2 and log N(O VIII(cm⁻²)) ≤ 14.5−15.2. An analysis of the data obtained with the combination of the Low Energy Transmission Grating (LETG) and the High Resolution Camera (HRC) on-board Chandra yielded consistent results. However, the data obtained with the LETG, combined with the Advanced CCD Imaging Spectrometer (ACIS) lead to the detection of an feature resembling an absorption line at λ ≈ 20 Å at simple one-parameter confidence level of 3.7σ, consistent with several earlier LETG/ACIS reports. Given the high statistical quality of the RGS1 data, the possibility of RGS1 accidentally missing the true line at λ ∼ 20 Å is very low: 0.006%. Neglecting this, the LETG/ACIS detection can be interpreted as Lyα transition of O VIII at one of the redshifts (z ≈ 0.054) of FUV-detected warm WHIM. Given the very convincing X-ray spectral evidence for and against the existence of the λ ∼ 20 Å feature, we cannot conclude whether or not it is a true astrophysical absorption line. Considering cosmological simulations, the probability of the LETG/ACIS λ ∼ 20 Å feature being due to the astrophysical O VIII absorber co-located with the FUV-detected O VI absorber is at the very low level of ≲0.1%. We cannot completely rule out the very unlikely possibility that the LETG/ACIS 20 Å feature is due to a transient event located close to the blazar.