ESA backpayment

Hello there this is my first post and I'm looking for a bit of help. I received a letter this week to apply for maybe a backpayment of ESA that was due to me. This is because I was transferred straight over to contribution based ESA instead of the Income based ESA which I was entitled too. After going through old bank statements I have realised that actually I may have been underpaid from 2012 to 2016. Now the form I received said it only related to 2012/13 when I was initially transferred from Incapacity Benefit. So my question is this. Is it possible that the DWP have after the initial year of me being transferred to contribution based ESA made a mistake and not then after the first year transferred me to the Income based ESA, but instead just kept me on Contribution based? Any help would be most appreciated..