I don't believe they could given the situation you described.
Not a legal expert but an enthusiast with a couple of trademarks.
Some thoughts that jump to mind:
You'd have to be doing something that p's them off.
If you aren't passing off their brand (pretending to be them) or infringing on their trademark (calling yourself something the same or confusingly similar) then that helps you a lot.
If you are taking potential business from them they might be p'd off!
But when you say organically this doesn't necessarily mean naturally. If you have links around the web with anchor text 'trademark' pointing to a page with affiliate links to 'trademark' business, they probably aren't going to like that.
If affiliate stuff is involved in the equation they'll probably disable your aff-id.
They could do things like ask you to not reference them, removing all trademarks (because you are p'ing them off taking traffic/sales from them). I don't think that you have to, but it depends on how you've been trying to rank organically and why?
Maybe this is a good example:
I go to Trafalgar Square and shout:
(1) Coca-Cola, Coca-Cola, Coca-Cola, Coca-Cola, Coca-Cola (1000's of times) *Coca-Cola can't really tell me to stop.
(2) Coca-Cola is a bad drink (1000 times)
*Coca-Cola can't really tell me to stop.
(3) Coca-Cola tastes like poo (1000 times)
*Coca-Cola may not have a right to stop me but they are going to keep an eye on me and the moment I make a mistake they'll have me
(4) Coca-Cola (1000 times) while selling another type of coke (now I'm passing off/infringing)
*Coca-Cola will get me to stop
If there is no context in how you are using John Lewis on your webpage, I'm presuming that there is a reason why you would need those visitors and possible to benefit from them, so I presume John Lewis may lose out because of it. If this is all correct then they may have a right to ask you to stop.
So basically, I have no idea what the answer is or if there is one without knowing more.
