We had a great time building this circuit with my colleagues, it was also a great challenge for them to tinker about the solution.
Once the solution was published, one of them proposed an even more elegant circuit twist !
We can make use of the fact that, when a transistor conducts, the collector also switches states !
So we can omit 1 of the relay contacts and rebuild this circuit as follows :
The circuit works the same way as the other one, except that the capacitor is discharged via the closed C-E path of T1 when conducting.
When T1 is blocking, C1 is charged via R3 and the current flowing through the relay coil.
R1 was also changed into 120 Ohm to reduce furthermore the Base current when C1 is discharging.