TPS61232: zero duty cycle mode

Part Number:TPS61232

We are currently finalizing our design where we are using a TPS630242 boost convert to generate 5V power rail from a single cell litium battery. I have 2 questions for it.

1. Will the 5V output remain stable when the input also is 5.0V?  IS this what zero duty cycle means- if the input voltage is higher than the target output voltage the output follows the input mius the Rds drop
2. We are missing the output discharge function, can the HYS pin be used for this if it is connected to Vout through a 100-200Ohm resistor - or is it simpler to just use the TPS61231?