Yes, we can use the compound branching logic and save the custom text in different custom variables based on the rank.

Step 1: Go to rank order question type >> Logic >> Compound branching >> Set the different criteria for all the answer options if they are selected as rank 1 go to the first follow-up question and update the custom variable 1 with the respective text.

Screenshots:

https://prnt.sc/rBYhHkQO-cK3

https://prnt.sc/s1ViGfi8t2xg

Step 2: Go to the first follow-up question >> Logic >> Compound branching >> Set the different criteria for all the answer options if they selected rank 2 go to the second follow-up question and update the custom variable 2 with the respective text.

Screenshots:

https://prnt.sc/9J2L5wgFdEjI

https://prnt.sc/chutwzVFSd3_

Repeat the steps for the next follow-up questions as well and use the respective custom variable syntax to display the values.

Eg. For custom variable 1 - ${custom1}

For custom variable 2 - ${custom2} and so on.