Rational connectedness for groups of proper projective similitudes

For a quadratic form $\phi$ over a field of characteristic different from $2$, we study whether its group of proper projective similitudes ${\mathbf{P}\mathbf{S}\mathbf{i}\mathbf{m}}^{+}(\phi )$ is rationally connected (i.e. $R$-trivial). We obtain new sufficient conditions in terms of structure properties of $\phi$. We further provide new examples of quadratic forms $\phi$ belonging to a given power of the fundamental ideal in the Witt ring and such that ${\mathbf{P}\mathbf{S}\mathbf{i}\mathbf{m}}^{+}(\phi )$ is not rationally connected.