  Kokol and Stopar ($2023$) recently studied the exact region
$\Omega_{\phi,\rho}$ determined by Spearman's footrule $\phi$ and Spearman's
$\rho$ and derived a sharp lower, as well as a non-sharp upper bound for $\rho$
given $\phi$. Considering that the proofs for establishing these inequalities
are novel and interesting, but technically quite involved we here provide
alternative simpler proofs mainly building upon shuffles, symmetry, denseness
and mass shifting. As a by-product of these proofs we derive several additional
results on shuffle rearrangements and the interplay between diagonal copulas
and shuffles which are of independent interest. Moreover we finally show that
we can get closer to the (non-sharp) upper bound than established in the
literature so far.
