This is an increasing problem with regard to constipation area committees and so on, digging their heels in and trying everything to prevent FTTP installs.
A slight change in the wording you use for such installs can work wonders, instead of full fibre internet install from open reach, referring to it as a new phone line, or replacing/renewing the phone line with some work by British Telecom (formerly known as the GPO) can help. If they ask for more details you can also add a small discreet junction box will need to be added
In my experience, a lot of these committees who rule Conservation areas are usually made up of older people who see the internet as a frivolity and certainly dont see the need for faster internet when their sub 2mb ADSL connection does the job for them just fine.
Gather some evidence, look round your conservation area and see what other buildings/properties have the grey openreach boxes and ( from a public area ie pavement) take photos to use in evidence
But refer to it as needing to replace the phone line - which technically it is as you can use it for Voip phone services as if you are in a fibre priority area thats the only option for voice calls
