We introduce the roots of supercritical
Leung--Yau--Zaslow pairs and prove pointwise strong positivity of the
associated Schur forms.
The resulting LYZ cones
contain the pliant cone on smooth projective threefolds, with
strict inclusion on a Fujino--Sato threefold. We also establish
Hard Lefschetz and Hodge--Riemann relations for the associated
Chern classes on compact K\"ahler manifolds in bidegrees
$1\le p+q\le\dim X$ with $\min\{p,q\}\le1$.