PNP array TD62783 output is not closing correctly when input signal goes down
There are quite a few issues with your circuit, but the main one for me is how can it be working in any way at all. 


The TD62783 is a poor choice for a LED driver, as it is a Darlington emitter-follower configuration. In circuits where I have seen this done, the large voltage drop and output impedance leads to low and variable intensity displays. 


You, however, are using a 12V supply, so this drop should not be too much of an issue, as it is in the 5V circuits I have seen. 


You are linearly driving individual LEDs off of 12V. That’s a lot of wasted power. 


LEDs forward voltage is less than 3V, every single volt above that is just wasted power. 


You are using 12V to drive individual LEDs without any current limiting. This would bring out the magic smoke in any LED and those drivers. 


The TD62783 can push more than 500mA, at 12V this would be 6W between a LED and the driver, and that is for just one channel. Popcorn comes to mind, however:


The TD62783 is an open-emitter driver, it does not actively drive a zero. 


This is why nothing has blown up yet. You are trying to pull currents out of the LED cathodes with a device that cannot pull current at all. 

So, the real question in my mind is what can possibly be pulling that node down at all, how can these be going down so fast, and how are those LEDs lighting up?

The only possibility I see is leakage currents, enough leakage to pull 10V down to zero in 1ms. Assuming 1000pF in the LED nodes, that would be ~10uA. Which would not be unreasonable for the back-polarized protection diodes of the TD62783 (and less than the 100uA maximum specified in the datasheet). But this should be nearly unnoticeable in an LED.