I have played some Sly3 today and made some testings and I have a theory. I think, this is my theory, it is faster to shake palm trees then just make him chase you

Can you send video?

Unfortunately I do not have a capture card, but I can try to film it with my mobile tomorrow

Here it is

Stone Jake takes a long and obnoxious path at times when using the palm trees. In the Any% run we just get caught by him and get him to run all the way to the end with our optimal line of movement. In the MTC (Patch Grab) we try to get his attention by whacking some trees then getting caught by him and doing any% strats.