Cornering the Planck $A_{lens}$ tension with future CMB data
  The precise measurements of Cosmic Microwave Background Anisotropy angular
power spectra made by the Planck satellite show an anomalous value for the
lensing amplitude, defined by the parameter $A_{lens}$, at more than $2$
standard deviations. In this paper, after discussing the current status of the
anomaly, we quantify the potential of future CMB measurements in
confirming/falsifying the $A_{lens}$ tension. We find that a space-based
experiment as LiteBIRD could falsify the current $A_{lens}$ tension at the
level of $5$ standard deviations. Similar constraints can be achieved by a
Stage-III experiment assuming an external prior on the reionization optical
depth of $\tau=0.055\pm0.010$ as already provided by the Planck satellite. A
Stage-IV experiment could further test the $A_{lens}$ tension at the level of
$10$ standard deviations. A comparison between temperature and polarization
measurements made at different frequencies could further identify possible
systematics responsible for $A_{lens}>1$. We show that, in the case of the
CMB-S4 experiment, polarization data alone will have the potential of
falsifying the current $A_{lens}$ anomaly at more than five standard deviation
and to strongly bound its frequency dependence. We also evaluate the future
constraints on a possible scale dependence for $A_{lens}$.
