  Let $(X,\omega)$ be a compact K\"ahler manifold and $\mathcal H$ the space of
K\"ahler metrics cohomologous to $\omega$. If a cscK metric exists in $\mathcal
H$, we show that all finite energy minimizers of the extended K-energy are
smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and
the second author. As an immediate application, we obtain that existence of a
cscK metric in $\mathcal H$ implies J-properness of the K-energy, thus
confirming one direction of a conjecture of Tian. Exploiting this properness
result we prove that an ample line bundle $(X,L)$ admitting a cscK metric in
$c_1(L)$ is $K$-polystable.
