First they need to complete the sale of the club – but as soon as the ink is dried on their takeover, Rudiger’s future needs to be secured.
The final score was 4-2 – but it would not have flattered the visitors if they had left Stamford Bridge with an even greater margin of victory.
In the absence of Rudiger – who was missing with a groin strain – Chelsea’s usually rock solid rear-guard was ragged and horribly exposed.
They gifted Arsenal three goals and were cut open at will.
While the nature of defeat cannot be solely attributed to the Rudiger-shaped hole in a Chelsea defence that was also without Thiago Silva from the start, it underlined the importance of the out of contract centre-back.
Meanwhile, another defender, who is also a free agent in Andreas Christensen, produced a latest error-strewn performance that will only lead to accusations that his head is already in Barcelona, where he is bound at the end of the season.
Tuchel does not get a free pass, either. His decision to name a makeshift back three of Reece James, Christensen and Malang Sarr left Chelsea there for the taking – and Arsenal relished the opportunity to capitalise.
At the other end, Romelu Lukaku – handed his first start since February 19 – effectively ensured he will be back on the bench against West Ham on Sunday.
He managed just one shot – off target – in 60 minutes of action and provided little threat to an Arsenal defence that has been taken apart by Crystal Palace and Brighton in recent weeks.
But this was about Chelsea’s back line, which undermined any hope of cementing their position in third place.
Five points separate them from Tottenham and Arsenal, even if they have a game in hand on their London rivals.
Playing like this, however, and Tuchel’s pre-match claim that third place could not be considered safe, looks a sound assessment.
True, Chelsea could be forgiven for being fatigued after their exploits in Madrid last week and an FA Cup semi-final at Wembley on Sunday. But there is little excuse for the individual errors that gifted the points to Arsenal, who were staring down the barrel of four straight league defeats for the first time since 1995.
Mikel Arteta’s side could hardly have been handed more of an invitation to take the points here.
A break-neck first 45 minutes may have ended all square, with four goals shared between the sides – but Arsenal will have known they should have further exploited the absence of Rudiger and Silva in the Chelsea starting XI.
Without that defensive duo, Tuchel’s defence looked clueless – and Arsenal’s attack, deadly.
The visitors were handed the opening goal when a woefully under-hit back-pass from Christensen – who was promptly hooked at half time – handed Eddie Nketiah his first of the night after 13 minutes.
Racing through on goal, he swept a shot past the advancing Edouard Mendy, who had already denied Gabriel Magalhaes early on.
Chelsea were level through Timo Werner four minutes later when the German’s shot from distance took a heavy deflection off Granit Xhaka.
Emile Smith Rowe restored Arsenal’s lead in the 26th minute after a slick break, which was aided by Marcos Alonso completely vacating his position when going in search of a goal himself.
Bukayo Saka and Martin Odegaard combined to set up Smith Rowe for a smart finish.
Chelsea responded again, with Mason Mount’s whipped cross flicked in at the near post by Cesar Azpilicueta after 32 minutes.
Arsenal continued to create the better of the chances before the break, but their finishing let them down.
Tuchel needed to make a change and did so when sending on Silva for Christensen at the start of the second-half.
But it was another defensive lapse – this time via a mix up between Malang Sarr and N’Golo Kante – that gifted Arsenal the lead for the third time 12 minutes into the half.
It was Nketiah who benefitted again – finishing from close range.
Tuchel threw on Kai Havertz and Hakim Ziyech as he went in search of another equaliser that never looked like coming.
There was still time for further humiliation when Azpilicueta brought down Saka late on in the box.
The Arsenal forward stepped up to convert the spot kick to complete Chelsea’s misery.