  The joint observation of GW170817 and GRB170817A proved that binary neutron
star (BNS) mergers are progenitors of short Gamma-ray Bursts (SGRB): this
established a direct link between the still unsettled SGRB central engine and
the outcome of BNS mergers, whose nature depends on the equation of state (EOS)
and on the masses of the NSs. We propose a novel method to probe the central
engine of SGRBs based on this link. We produce an extended catalog of BNS
mergers by combining recent theoretically predicted BNS merger rate as a
function of redshift and the NS mass distribution inferred from measurements of
Galactic BNSs. We use this catalog to predict the number of BNS systems ending
as magnetars (stable or Supramassive NS) or BHs (formed promptly or after the
collapse of a hypermassive NS) for different EOSs, and we compare these
outcomes with the observed rate of SGRBs. Despite the uncertainties mainly
related to the poor knowledge of the SGRB jet structure, we find that for most
EOSs the rate of magnetars produced after BNS mergers is sufficient to power
all the SGRBs, while scenarios with only BHs as possible central engine seems
to be disfavoured.
