[Oooops, I was still expanding on my own post when you popped this question]They might, but to equate that action with 'intent' then they'd need to provide a basis for doing so. The Case Law doesn't quite cover that action. Some argument about the cost of travel on each leg of the journey, and on the operation of the machine (the Oyster touch pad) would appear to be necessary. I'd be uncomfortable in having to argue that from the Company's position, and I can't say I'd cherish arguing it from the passenger's position but its I'd be marginally more confident with the passenger's defence.
But I still won't estimate the odds, and without those, I couldn't advise on the costs of legal representation.
Anyway, I will say that if the passenger WAS to reply themselves (and therefore to proceed without representation), then I'd suggest making a brief but clear statement of the health matter, a confident assertion that there was no intention to avoid the fare, an honest explanation of the reason for not paying, and refute that passing the ticket (Oyster Card) over the touch-pad demonstrates anything other than compliance with fare paying procedure.
To go back to the letter that is currently waiting a response. I should add that it SHOULD be responded to, but if the passenger IS instructing a Solicitor, then the Solicitor should advise. However, even when it has been sent, there is still the possibility to negotiate with the Company between submitting the Statement and any possible Court Action. In fact a great many cases are negotiated between the submission of Evidence and a Court Hearing, for the simple reason that it can save time and money - it can be cheaper for the Company to accept a settlement or to abandon a doomed Claim, it can be quicker, and its advantageous for the passenger to conclude the matter sooner, perhaps at less cost, avoid the inconvenience of a Court appearance and avoid the risk of a Prosecution.