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But now, as he prepares to hand over the Anfield reins, the issue has once again become a cause for concern.
The Reds' indifferent record against their main rivals has been one of the nagging doubts throughout a campaign in which their Premier League title challenge fizzled out during the past month.
Many have traced the turning point back to the draw at Manchester United at the start of April, when Klopp's side dominated the hosts during a one-sided first half but ultimately needed a late Mohamed Salah penalty to snatch a 2-2 draw.
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There had been a similar story earlier in the campaign at Anfield when a succession of missed opportunities allowed United to escape down the other end of the East Lancs Road with a barely-merited point from a goalless stalemate.
That United also eliminated Liverpool from the FA Cup in March in similar confusing fashion with Klopp's side having been by some distance the better team further exposed the issue.
For whatever reasons, the Reds have too often not been able to put away their traditional top-six opponents.
Yes, there were notable victories at Arsenal in the FA Cup third round and, of course, Chelsea in the League Cup final.
But in terms of Premier League matches, Liverpool's win over Tottenham Hotspur last Sunday meant they have won only two of 10 games in the top flight this term against Manchester sides City and United and London trio Arsenal, Chelsea and Tottenham.
The Reds have taken just 12 points from those games to stand fourth in the mini-league of the six clubs this season.
But Arsenal are the clear leaders, unbeaten in nine games and on 19 points with a trip to Old Trafford to come at the weekend.
Only once under Klopp have Liverpool taken fewer points from such matches, when mustering a mere 10 in the 2017/18 campaign.
Last season, though, they slipped to fourth having taken just 15 points, with their recent away record seeing only one away win in their last 11 Premier League trips to their traditional top-six rivals, the 2-1 triumph at Tottenham in November 2022.
Given that the league winners in seven of Klopp's full seasons in charge have finished in the top two of that mini-league shows where Liverpool have ultimately fallen short this term.
Of course, Newcastle United and Aston Villa would argue the top six has become a top eight in the last two years.
And, curiously, if points were restricted to just games between the current top eight, the Reds would be only a point behind leaders Arsenal.
But the shortcoming against top clubs is nevertheless glaringly apparent.
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