Here is a "delay off timer" circuit that should work (I didn't breadboard it yet).
The relay energizes immediately when any Pin-16 input goes low.
When all Pin-16 inputs then open, the relay will remain energized for the duration of the timer setting. The timer range is approx 30-60 seconds.
How it works....assume 5v,12v power is applied to the circut.
When all P16 inputs are high:
Q1 is off and Timer is off, so Q3 is off, and therefore, the relay is off.
Q2 is off, therefore, no power is applied to the 555.
When any P16 input goes low:
Q1 turns on and turns on Q3. The relay immediately turn on.
Q3 also turns on Q2. Q2 applies power to the timer, so the timer turns on.
However, the timer does not start because "Tr" is at 5v (C3 is not discharged).
When the last remaining P16 input goes from low to high:
Q1 turns off and would turn off Q3. However, a negative going pulse is generated by
C3 that immediately triggers the timer and keeps Q3 on.
The timer starts and when the time expires, Q3 turn off. The relay deenergizes.
Q3 also turns off Q2 and removes power from the Timer.
The circuit is now reset for the next cycle.
Hope this makes sense. Like I said, I haven't tested this so its still "theory".