Possibly increased to 470k, assuming that all 10 switches are closed, the total resistance of all 1 meg resistors in parallel would be 100k, so the voltage at pin 2 would be 12v * 100k/570k = 2.7 V( including diode vdrop)that would be low enough to trigger the timer.
Try it with ten switches, and see, increasing the 100k to 470 to 680k should be fine.
Try it with ten switches, and see, increasing the 100k to 470 to 680k should be fine.