  In 1988 Kalai construct a large class of simplicial spheres, called squeezed
spheres, and in 1991 presented a conjectured about generic initial ideals of
Stanley--Reisner ideals of squeezed spheres. In the present paper this
conjecture will be proved. In order to prove Kalai's conjecture, based on the
fact that every squeezed $(d-1)$-sphere is the boundary of a certain $d$-ball,
called a squeezed $d$-ball, generic initial ideals of Stanley--Reisner ideals
of squeezed balls will be determined. In addition, generic initial ideals of
exterior face ideals of squeezed balls are determined. On the other hand, we
study the squeezing operation, which assigns to each Gorenstein* complex
$\Gamma$ having the weak Lefschetz property a squeezed sphere
$\mathrm{Sq}(\Gamma)$, and show that this operation increases graded Betti
numbers.
