Liverpool can lose it's first position only if they show worst performance in there upcoming matches, which is not possible because Liverpool has lost just one game this season. Even if Liverpool lose 6 or more of it's remaining matches then Arsenal has to win almost all of it's remaining games, which won't be a piece of cake.
One main reason why Liverpool is Champions-Elect is that they are now out from all other leagues and there sole focus is now on winning EPL title. Arsenal once again has to bear second position in EPL if they play well in remaining games. The points gap between Arsenal and Nottingham is of just 4 points.
It's almost impossible because Liverpool will probably never lose consecutive matches in their next one so at this stage I think they will never be knocked off the top of the Premier League. On the other hand it is now clear as water to us that Arsenal will never be able to top the EPL. Moreover according to the conversation you mentioned, Liverpool Football Club's sole goal is to win the EPL trophy. The truth is that, Liverpool Football Club may have given their best in everything before but due to some of their weak points they now have to constantly plan to fight differently to achieve the highest status. By the way Arsenal will definitely play well in the remaining games and they will face Fulham.
From remaining nine games Liverpool needed only 3 wins for cementing their title race which is looking not challenging for them due to their recent performance and consistency which give them huge lead at the top now they are involved only in Premier League which is also good advantage for them.
Sad exit of Champions League and stunning defeat in EFL Cup is surely going to have impact on their performance but still things are balanced for them to hold their nerves and stay at the top of league table Arsenal is now having good chance of keeping themselves at the second spot while Nottingham Forest is also set for third spot. In this season, Premier League have 5 spots in Champions League which will helpful for them, but things are looking competitiveness as they are having few toughest teams to fight hard for these spots.