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\section{Results}  \subsection{RGB}  Our calculations confirm the results of \citet{Eggenberger:2012}. Models that only include angular momentum transport due to rotational instabilities and circulations fail to reproduce the splittings  observed splittings. in the RGB star $\KIC$.  The resulting cores are so rapidly rotating that the the  perturbative approach to the splitting calculation is no more justified. This said, even Even  models with an extremely slow initial rotation of $1 \kms$ result in rotational splittings 1 one  order of magnitude larger than the observed ones. ones, which clearly shows this class of models can not explain the observations.  This is in perfect agreement with the calculations of \citet{Eggenberger:2012}, even if the implementation of the physics of rotation is quite different in the GENEVA code compared to MESA \citep[See Sec.6 in ][and references therein]{Paxton:2013}.Models that include angular momentum transport due to Spruit-Tayler magnetic fields \citep[which are generated by differential rotation in radiative regions][]{Spruit:2002}   do a much better job, but still result in rotational splittings on the order of $1\mu$Hz. We explored if an increase in the efficiency of the ST mechanism could reconcile the models with the observations. However even increasing the efficiency of the diffusion coefficient resulting from the magnetic torques by a factor 100 result in a very small change in the overall coupling. This is because the poloidal component of the magnetic field is generated by the Tayler instability in the toroidal component. However the toroidal component results from differential rotation and differential rotation is suppressed by the presence of a radial component of the magnetic field, the sy is not  The reason We found that models including also angular momentum transport due to Spruit-Tayler magnetic fields \citep[which are generated by differential rotation in radiative regions][]{Spruit:2002} do a much better job, but still result in rotational splittings on the order of $1\mu$Hz. We explored if an increase in the efficiency of the ST mechanism could reconcile the models with the observations. However even increasing the efficiency of the diffusion coefficient resulting from the magnetic torques by a factor 100 result in a very small change in the overall coupling. This  is due to  the self-regulating nature of the Spruit-Tayler dynamo. The poloidal component of the magnetic field $B_r$ is generated by the Tayler instability that occurs in the toroidal component $B_{\phi}$. However the toroidal component is amplified by the differential rotation, which is in turn suppressed by the torque $\propto B_r B_{\phi}$. It is not too surprising then to observe that the system tend to relax around some average differential rotation state which is weakly dependent on a possible increase of the efficiency of the Tayler-Spruit  dynamo loop: s loop.     We recall that, while the physics of the Tayler instability is solid, the existence of the Tayler-Spruit dynamo loop is currently theoretically debated \citep{Braithwaite:2006,Zahn:2007}. From the observational point of view observations of the spin rates of compact objects (WD and NS) are in much better agreement with models including this angular momentum transport mechanism \citep{Heger:2005,larends_Yoon_Heger_Herwig_2008}, which has also been discussed in the context of the rigid rotation of the solar core \citep{Eggenberger:2005}, but see also \citet{Denissenkov:2010}