It’s hitting the wrong target. Putin won’t care what the Sun says about a fake story.
He will. He's not going to be happy about this being all over the internet, whether it's bollocks or not.
We know this because of the way he's taken such care to portray himself as a strongman, in the physical sense of the term. All those topless photos hunting etc. That may also be the reason why he's gone so heavy on the Botox of late too. It's important to him to be regarded as fit, healthy and vital. And that becomes more of an issue as he ages. Also, dictators pretty much always resent having the piss taken out of them. They understand that humour is a potent weapon, which is why they're a pretty thin skinned bunch.
Obviously people can decide for themselves if they feel comfortable with it or not. That's a personal preference. But it's clear how he's going to feel about this, because of the threat it poses to the narrative he has tried so hard to create.