In message
p.homeip.net, Paul
Cummins writes

You don't need an OEP to Twickenham - it's within the Oyster area.

Being within the Oyster area is irrelevant. In fact OEPs have no
validity at all outside the Oyster area.

You only need an OEP if you have a non PAYG oyster and are travelling
outside your validity with it.

No. They are needed when travelling on National Rail and using PAYG to
go beyond the validity of an Oyster travelcard.

As has been pointed out, the correct fare will be charged on touching
out, even if there is no OEP, but the user is technically liable for a
penalty fare if their Oyster is gripped en route while outside of their
zones.

--
Paul Terry