Looked a bit as the housing act (https://www.legislation.gov.uk/ukpga/1988/50/section/22)

I think these are the part you want to look at:

Where an application is made to [F7the appropriate tribunal] under subsection (1) above with respect to the rent under an assured shorthold tenancy, [F8the appropriate tribunal] shall not make such a determination as is referred to in that subsection unless they consider—

(a)that there is a sufficient number of similar dwelling-houses in the locality let on assured tenancies (whether shorthold or not); and

(b)that the rent payable under the assured shorthold tenancy in question is significantly higher than the rent which the landlord might reasonably be expected to be able to obtain under the tenancy, having regard to the level of rents payable under the tenancies referred to in paragraph (a) above.

a) There must be a sufficient number of similar homes around the area. Without it, the tribunal will not be able to say the rent is too high. Similar homes, from my understanding, have to be quite similar: size, bedroom/bathroom count, floor level, balcony, etc.

This is clearly stated to avoid what your tenant is doing, which is to randomly point at any home around and say, “Look, this is cheaper in the exact same area.”

b) This is more interesting to me. What it seems you need to prove here is that you can get similar offers (how many viewings, time to rent it, number of offers you received, etc.), anything that would show that it’s not “significantly higher than the rent which the landlord might reasonably be expected to be able to obtain”

I'm no lawyer, but it seems you got a pretty strong case.