I don't know when the Vlahovic links first emerged, but I found this.Lends credence to your take as a possible explanation. Is this around the same time as our Vlahovic links?
1 July 2021
With the guidance of Paratici and Espirito Santo, Tottenham have opened talks over a move for 21-year-old Fiorentina striker Dusan Vlahovic.

Dusan Vlahovic could becomes Nuno Espirito Santo's first signing as Spurs boss
Spurs' new director of football Fabio Paratici has welcomed Nuno Espirito Santo to Tottenham and has already eyed his first signing, with the club looking to recruit this summer
Yes, he can. And Conte described Sonny as a "modern player", so it would seem he likes players that can offer a little bit of dynamic movement.
Dunno. Maybe it's my poor English, but he seemed to be suggesting yesterday that he prefers Kane over Salah. Salah can be decisive in any game, you see, but Kane makes the difference in training.In light of your theory, perhaps Conte has a similar misunderstanding with Kane and doesn't appreciate all that he brings.
“Harry is a world-class striker. Every day that I train him I understand very well that he's a world-class striker. He's one of the best I've had the pleasure to train. He's amazing."
"I like to have him in my team. For sure Jurgen likes to have Mo Salah. But I like to have Harry in my team. He's an important player for us and my expectation was very high and to play an important game against Liverpool.
“Every time [Salah] has the ball he’s a danger. He improved a lot, I remember him when he played in Italy.
"Now he’s improved in an incredible way. Now I think he’s one of the best players in the world and in every game he’s very decisive.”

Conte makes "very strange" Kane admission amid slump in Tottenham form
Harry Kane failed to secure a Tottenham exit in the summer and has struggled to be his usual prolific self which has led Antonio Conte to comment on his recent form
Dunno, but according to The Athletic, preseason training and an open transfer window give Conte the horn like nothing else. You can imagine how he might have turned up his nose at our lastminute.com MO. How can you prepare in those conditions?This could go either way but to be consistent with your take it would mean Conte wanted to see if Kane was playing the leverage game to, 1) get more money or 2) get DL to spend...or really leave. I can see a manager not wanting the uncertainty. Kane coming back seemed to affect performances somewhat.
Dunno.