  Short gamma-ray bursts (SGRBs) are among the most luminous explosions in the
Universe and their origin still remains uncertain. Observational evidence
favors the association with binary neutron star or neutron star-black hole
(NS-BH) binary mergers. Leading models relate SGRBs to a relativistic jet
launched by the BH-torus system resulting from the merger. However, recent
observations have revealed a large fraction of SGRB events accompanied by X-ray
afterglows with durations $\sim10^2\!-\!10^5~\mathrm{s}$, suggesting continuous
energy injection from a long-lived central engine, which is incompatible with
the short ($\lesssim1~\mathrm{s}$) accretion timescale of a BH-torus system.
The formation of a supramassive NS, resisting the collapse on much longer
spin-down timescales, can explain these afterglow durations, but leaves serious
doubts on whether a relativistic jet can be launched at merger. Here we present
a novel scenario accommodating both aspects, where the SGRB is produced after
the collapse of a supramassive NS. Early differential rotation and subsequent
spin-down emission generate an optically thick environment around the NS
consisting of a photon-pair nebula and an outer shell of baryon-loaded ejecta.
While the jet easily drills through this environment, spin-down radiation
diffuses outwards on much longer timescales and accumulates a delay that allows
the SGRB to be observed before (part of) the long-lasting X-ray signal. By
analyzing diffusion timescales for a wide range of physical parameters, we find
delays that can generally reach $\sim10^5~\mathrm{s}$, compatible with
observations. The success of this fundamental test makes this "time-reversal"
scenario an attractive alternative to current SGRB models.
