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Since the publication of the results of the Planck satellite mission in 2013, the local and early universes have been considered to be in tension in respect of the determination of amplitude of the matter density spatial fluctuations ($\sigma_8$) and the amount of matter present in the universe ($\Omega_m$). This tension can be seen as a lack of massive galaxy clusters in the local universe compared to the prediction inferred from Planck cosmic microwave background (CMB) best-fitting cosmology. In the present analysis, we perform the first detection of the cross-correlation between X-rays and CMB weak-lensing at 9.1 $\sigma$. We next combine thermal Sunyaev-Zel'dovich, X-rays, and weak-lensing angular auto and cross power spectra to determine the galaxy cluster hydrostatic mass bias. We derive $(1-b_H) = 0.70 \pm 0.05$. Considering these constraints, we observe that estimations of $\sigma_8$ in the local Universe are consistent with Planck CMB best-fitting cosmology. However, these results are in clear tension with the output of hydrodynamical simulations that favor $(1-b_H) > 0.8$.

First measurement of the cross-correlation between CMB weak lensing and X-ray emission