  The nature of solar wind (SW) turbulence below the proton gyroscale is a
topic that is being investigated extensively nowadays. Although recent
observations gave evidence of the dominance of Kinetic Alfv\'en Waves (KAW) at
sub-ion scales with $\omega<{\omega_{ci}}$, other studies suggest that the KAW
mode cannot carry the turbulence cascade down to electron scales and that the
whistler mode (i.e., $\omega>\omega_{ci}$) is more relevant. Here, we propose
to study key properties of the short wavelength plasma modes under realistic SW
conditions, typically $\beta_i\gtrsim \beta_e\sim 1$ and for high oblique
angles of propagation $80^\circ\leq \Theta_{\bf kB}<90^\circ$ as observed from
the Cluster data. The linear properties of the plasma modes under these
conditions are poorly known, which contrasts with the well-documented cold
plasma limit and/or moderate oblique angles of propagation ($\Theta_{\bf kB}
<80^\circ$). Based on linear solutions of the Vlasov kinetic theory, we discuss
the relevance of each plasma mode (fast, Bernstein, KAW, whistler) in carrying
the energy cascade down to electron scales. We show, in particular, that the
shear Alfv\'en mode extends at scales $k\rho_i\gtrsim1$ following either a
whistler mode ($\omega>\omega_{ci}$) or a KAW mode (with $\omega<\omega_{ci}$)
depending on the anisotropy $k_\parallel/ k_\perp$. This contrasts with the
well-accepted idea that the whistler branch develops as a continuation at high
frequencies of the fast magnetosonic mode. We show, furthermore, that the
whistler branch is more damped than the KAW one, which makes the latter a more
relevant candidate to carry the energy cascade down to electron scales. We
discuss how these new findings may facilitate resolution of the controversy
concerning the nature of the small scale turbulence, and we discuss the
implications for present and future spacecraft wave measurements in the SW.
