  In standard cosmology, the late Universe is assumed to be statistically
homogeneous and isotropic. However, a recent study based on galaxy clusters by
Migkas et al. (2021, arXiv:2103.13904) found an apparent spatial variation of
approximately $9\%$ in the Hubble constant, $H_0$, across the sky. The authors
utilised galaxy cluster scaling relations between various cosmology-dependent
cluster properties and a cosmology-independent property, i.e., the temperature
of the intracluster gas $(T)$. A position-dependent systematic bias of $T$
measurements can, in principle, result in an overestimation of apparent $H_0$
variations. In this study, we search for directional $T$ measurement biases by
examining the scaling relation between the member galaxy velocity dispersion
and the gas temperature $(\sigma_\mathrm{v}-T)$. Additionally, we search for
apparent $H_0$ angular variations independently of $T$ by analysing the
relations between the X-ray luminosity and Sunyaev-Zeldovich signal with the
velocity dispersion, $L_\mathrm{X}-\sigma_\mathrm{v}$ and
$Y_\mathrm{SZ}-\sigma_\mathrm{v}$. We utilise Monte Carlo simulations of
isotropic cluster samples to quantify the statistical significance of any
observed anisotropies. We find no significant directional $T$ measurement
biases, and the probability that a directional $T$ bias causes the previously
observed $H_0$ anisotropy is only $0.002\%$. On the other hand, from the joint
analysis of the $L_\mathrm{X}-\sigma_\mathrm{v}$ and
$Y_\mathrm{SZ}-\sigma_\mathrm{v}$ relations, the maximum variation of $H_0$ is
found in the direction of $(295^\circ\pm71^\circ, -30^\circ\pm71^\circ)$ with a
statistical significance of $3.64\sigma$, fully consistent with
arXiv:2103.13904. Our findings strongly corroborate the previously detected
spatial anisotropy of galaxy cluster scaling relations using a new independent
cluster property, $\sigma_\mathrm{v}$.
