Oxf & Bucks LI / 1st GreenJackets Cap?

[leigh kitchen]

This cap has no coloured piping and is the plain dark green cap of light infantry regiments if the detachable black mohair band is removed.
The buttons are the St Edwards Crown black horn or plastic ones of the Green Jacket Brigade and later the Royal Green Jackets (and now the Rifles?)They are sewn directly to the cap, not to the mohair band.

Given the combination of the lack of black piping to the cap, the presence of Green Jacket Brigade buttons and the black mohair band, and the caps date of manufacture, 1963, am I right in thinking that this is an Oxfordshire and Buckinghamshire Light Infantry cap "converted" to wear by the 1st Bn after they became the 1st Green Jackets (Regiment) within the Green Jackets Brigade (and prior to becoming the 1st Bn of the "new" regiment named the Royal Green Jackets)?
In which case it would bear the Green Jackets Brigade cap badge?

The cap is by J Compton Sons & Webb London, ink stamped on the sweatband is "1963" and written on it is a fairly distinctive name, "L/CPL FOZARD. G. (BAND)"