Q: Indexing of erasure code division in Availability and Validity correct?

Regarding the Availability and Validity paper from Handan Kilinc Alper, I was wondering whether the indexing for the proof in the Merkle tree construction for each parachain validator PV_i was indeed correct (page 3 top). To be more precise, instead of i in \pi_i^{mt}, shouldn’t it be j to proof that the code piece d_j is included in the Merkle tree?

I am sorry if I misunderstood something and look forward to your answer.