Here's the breakdown of the letters I've used when presenting this possibility:
After establishing the Corinthian church Paul writes the church a letter on the avoidance of immorality (I Cor. 5:9). This is either a lost letter or it could be II Cor. 6:14 - 7:1.
After receiving a letter from the Corinthian church with questions and he also learns some things from Chloe. He responds with what we call I Corinthians.
At some point after I Corinthians is sent, someone arrives in Corinth and stirs up opposition to Paul (II Cor. 11:13-15, 23). They have letters of recommendation (3:1). They were Jewish (11:22) but there's not indication they're from Jerusalem. These visitors made demands (11:19-21), scoffed at Paul (11:30), and challenged Paul's authority and inspiration (13:3). It seems one attack on Paul's legitimacy was that he (Paul) didn't ask for money from the church for himself (11:7-11; 12:13-18). It appears these visitors were good speakers (11:6), claimed to have visions and revelations of the Lord (12:1), but preached a different gospel from the one Paul preached (11:4)
Paul learns of this visit and writes a letter ( which could be =II Cor 2:14 - 6:13; 7:2 - 4)
Real danger to Paul's leadership emerges (10:10) so he visits Corinth finding them rebellious and he is insulted (2:5 - 11; 7:12)
Paul sends a furious letter--one he later says he hated having to send (7:8, 12; 2:1 - 4). (That letter = II Cor. 10 - 13).
Paul sends Titus to check on things (12:17) and learns that his correspondence has been successful. The church has repented and the one who insulted Paul has been censured (2:6; 7:5 - 11).
Paul writes a conciliatory letter (which is = II Cor 1:1 - 2:13; 7:5 - 9:15).
If you count the letter mentioned in I Cor. 5:9 as a lost one rather than one included within II Corinthians you come up with five possible letters. Again, an interesting conjecture.