OR would need you to sign a 'Permission to dig' form to carry out the work. If you don't sign it, they will find an alternative solution.
Shared cable bundles, conduit, O/H wiring ... are a legacy of the infrastructure put down decades ago. OR try to provide a stand-alone separate feed to premises, these days. If it was FTTP you'd have your own separate BFT provided.
Are you sure it's a shared conduit (or duct as we refer to it as) and not a shared bundle (underground armoured cable) ??
If your house was built in the 60/70's I suspect it is likely to be the latter as that was the preferred method back then. It is unusual (although not unheard of), for a fault to appear in a ducted cable ... only had it once myself and it was rats chewing on it.
Conversly, the armoured cables deteriorate badly and end up with all manner of faults on the circuits.
Just my two-penneth .... not a legal bod by any means but believe me when I say OR will
always work to secure a solution that fits all parties. Just be polite, there's nothing worse than some so-and-so screaming at you in colourful language that we won't be touching their property.