  We introduce the notion of essential support of a simple Gelfand-Tsetlin
$\mathfrak{gl}_n$-module as an important tool towards understanding the
character formula of such module. This support detects the weights in the
module having maximal possible Gelfand-Tsetlin multiplicities. Using
combinatorial tools we describe the essential supports of the simple socles of
the universal tableaux modules. We also prove that every simple Verma module
appears as a socle of a universal tableaux module and hence obtain a
description of the essential supports of all simple Verma modules. As a
consequence, we prove the Strong Futorny-Ovsienko Conjecture on the sharpness
of the upper bounds of the Gelfand-Tsetlin multiplicities. In addition we give
a very explicit description of the support and essential support of the simple
singular Verma module $M(-\rho)$
