The progenitor systems of Type-Ia supernovae ( SNe Ia ) are yet unknown .
The collisional-triple SN Ia progenitor model posits that SNe Ia result from head-on collisions of binary white dwarfs ( WDs ) , driven by dynamical perturbations by the tertiary stars in mild-hierarchical triple systems .
To reproduce the Galactic SN Ia rate , at least \sim 30 - 55 per cent of all WDs would need to be in triple systems of a specific architecture .
We test this scenario by searching the Gaia DR2 database for the postulated progenitor triples .
Within a volume out to 120 pc , we search around Gaia -resolved double WDs with projected separations up to 300 au , for physical tertiary companions at projected separations out to 9000 au .
At 120 pc , Gaia can detect faint low-mass tertiaries down to the bottom of the main sequence and to the coolest WDs .
Around 27 double WDs , we identify zero tertiaries at such separations , setting a 95 per cent confidence upper limit of 11 per cent on the fraction of binary WDs that are part of mild hierarchical triples of the kind required by the model .
As only a fraction ( likely \sim 10 per cent ) of all WDs are in < 300 au WD binaries , the potential collisional-triple progenitor population appears to be at least an order of magnitude ( and likely several ) smaller than required by the model .