  Since the very first observations, the Cosmic Microwave Background (CMB) has
revealed on large-scales unexpected features known as anomalies, which
challenge the standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmological
model. One such anomaly is the "lack-of-correlation", where the measured
two-point angular correlation function of CMB temperature anisotropies is
compatible with zero, differently from the predictions of the standard model.
This anomaly could indicate a deviation from the standard model, unknown
systematics, or simply a rare realization of the model itself. In this study,
we explore the possibility that the lack-of-correlation anomaly is a
consequence of living in a rare realization of the standard model, by
leveraging the potential information provided by the cosmological gravitational
wave background (CGWB) detectable by future gravitational wave (GW)
interferometers. We analyze both constrained and unconstrained realizations of
the CGWB to investigate the extent of information that GWs can offer. To
quantify the impact of the CGWB on the lack-of-correlation anomaly, we employ
established estimators and introduce a new estimator that addresses the
"look-elsewhere" effect. Additionally, we consider three different maximum
multipoles, denoted as $\ell_{\rm max}$, to account for the anticipated
capabilities of future GW detectors ($\ell_{\rm max} = 4, 6, 10$). Summarizing
our findings for the case of $\ell_{\rm max} = 4$, we identify the angular
range $[63^\circ - 180^\circ]$ as the region where future observations of the
CGWB maximize the probability of rejecting the standard model. Furthermore, we
calculate the expected significance of this observation, demonstrating that
98.81% (81.67%) of the GW realizations enhance the current significance of the
anomaly when considering the full-sky (masked) Planck SMICA map as our CMB sky.
