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We use the 2015 Planck likelihood in combination with the Bicep2/Kecklikelihood (BKP and BK14) to constrain the chirality, $\chi$, of primordialgravitational waves in a scale-invariant scenario. In this framework, theparameter $\chi$ enters theory always coupled to the tensor-to-scalar ratio,$r$, e.g. in combination of the form $\chi \cdot r$. Thus, the capability todetect $\chi$ critically depends on the value of $r$. We find that with presentdata set $\chi$ is \textit{de facto}unconstrained. We also provide forecastsfor $\chi$ from future CMB experiments, including COrE+, exploring severalfiducial values of $r$. We find that the current limit on $r$ is tight enoughto disfavor a neat detection of $\chi$. For example, in the unlikely case inwhich $r\sim0.1(0.05)$, the maximal chirality case, i.e. $\chi = \pm1$, couldbe detected with a significance of $\sim2.5(1.5)\sigma$ at best. We concludethat the two-point statistics at the basis of CMB likelihood functions iscurrently unable to constrain chirality and may only provide weak limits on$\chi$ in the most optimistic scenarios. Hence, it is crucial to investigatethe use of other observables, e.g. provided by higher order statistics, toconstrain these kind of parity violating theories with the CMB.

Testing chirality of primordial gravitational waves with Planck and future CMB data: no hope from angular power spectra