  Strong stability is a property of time integration schemes for ODEs that
preserve temporal monotonicity of solutions in arbitrary (inner product) norms.
It is proved that explicit Runge--Kutta schemes of order $p\in 4\mathbb{N}$
with $s=p$ stages for linear autonomous ODE systems are not strongly stable,
closing an open stability question from [Z.~Sun and C.-W.~Shu, SIAM J. Numer.
Anal. 57 (2019), 1158--1182]. Furthermore, for explicit Runge--Kutta methods of
order $p\in\mathbb{N}$ and $s>p$ stages, we prove several sufficient as well as
necessary conditions for strong stability. These conditions involve both the
stability function and the hypocoercivity index of the ODE system matrix. This
index is a structural property combining the Hermitian and skew-Hermitian part
of the system matrix.
