Your canonical headshot lives where you do not control it
For most senior people, the photograph that matters is not the one on their own profile. It is the one communications sent to a conference organiser three years ago, which is now on a speaker page, an agenda PDF, a press release, two trade publications that covered the event and an archive copy of all of the above.
That file is high resolution, frontal, professionally lit and published in multiple independent places, which makes it close to ideal as an index anchor. It is also, in practice, unretractable. Any plan that begins with changing your LinkedIn photo has not addressed the actual exposure.