  We establish an algebraic approach to prove the properness of moduli spaces
of K-polystable Fano varieties and reduce the problem to a conjecture on
destabilizations of K-unstable Fano varieties. Specifically, we prove that if
the stability threshold of every K-unstable Fano variety is computed by a
divisorial valuation, then such K-moduli spaces are proper. The argument relies
on studying certain optimal destabilizing test configurations and constructing
a Theta-stratification on the moduli stack of Fano varieties.
