Okay...here is my related experience. Bear in mind I was flying Lufthansa and not the airlines you mentioned.
In September 2017 I toured from Dresden to Budapest. All of my flights were on Lufthansa operated aircraft. I THINK, but cannot recall for certain, that I booked through Lufthansa's USA website. Similar to what you said, there was a requirement on the website's sports equipment page that I contact Lufthansa customer service prior to my flight. I contacted as soon as I saw this, and did not wait until shortly before my flight.
This is what the customer service rep told me:

• The purpose of calling customer service in advance is because they wanted to reserve space for the bike. I was told Lufthansa aircraft are equipped with on-board racks for securing bicycles. I was also told that since all of my flights were on Lufthansa operated aircraft, I was NOT to box the bike. She also said that nonboxing would NOT be available if I was using a code share partner airline on any part of my itinerary. She said that the airline would accept a boxed bike if I wanted to do that, but it was preferred they not be boxed since it is easier and more efficient for the baggage handlers to roll the bike around than to have to move heavy, bulky boxes. She did not say this, but I gathered any bikes exceeding available rack space would have to be boxed. The customer service agent reserved rack space for the unboxed bike on all of my flights. For the record, when I was changing planes in Munich, I was able to see inside the hold of the connecting Dresden bound flight and could see my bike secured on the wall in the aircraft hold. I thought that was really cool.
• The agent asked if I was checking any other baggage, which I responded no as all my other gear is in a carry-on duffel. She then said that since the bicycle was the only checked baggage, the bicycle fee would be waived. Naturally I got her name and noted the time and date of my call.

Now the cautionary part. Having a fair deal of experience flying with a bike, I know the ultimate call is made at the departure airport. Three days before my flight, I went to the airport and waited for the Lufthansa agents to process all passengers for that day's flight. There is only one Lufthansa flight a day from my local airport, so I let them process that flight's passengers before interrupting their time to answer my questions. Once they were available, I asked them about the same information I had received from Lufthansa customer service. They knew nothing about it and said there would be a fee and the bike would have to be boxed. Eventually, they called over a manager (the only one with a German accent), and she confirmed the customer service rep's statements, though it seemed she did so reluctantly. Later that day, I called Lufthansa customer service again and told them there seemed to be a little confusion at the airport regarding the boxing and fee requirement. (I did this because the German accented manager told me she would not be working on the day of my flight, and I was concerned that day's manager may not know the policy.) Customer service was quite adamant they did not want the bike boxed, so I felt a little uncomfortable as to how things may play out on the day of my flight. Ultimately, I arrived at the airport early with an unboxed bike, but my wife waited at a cell phone lot in our car with a box should I need it. As luck would have it, the German gate agent's schedule changed and she was there, remembered me, and all I had to do was let the air out of the tires.

Now after all of that long story, my point is that the purpose of calling the airlines may be a similar situation where they want to ensure bicycle space is available. I recommend calling asap. However, get the name, contact information and time/date of your call to customer service as you may need it when you check in for your flights.

Now, as I say to anyone flying with their bikes.......good luck. You may need it.