K&K show a brass rose badge for this school. It did not have an OTC nor is any Cadet Corps listed in the 1930's Army Lists and the badge shown is perfect match for a Lancs collar badge.
The school certainly had a cadet unit in WW2,
https://collections.hampshireculture...oke-army-cadet
It is not clear why they would not be wearing the Hants badge like all other ACFs in the county.
A slidered brass rose badge exists but it is unattributed outside K&K. A white metal version has been attibuted to Hampshire County Cadets which the school would have been a part of. It's possible the sldiered badge dates from 1930-2 when the cadet units were not allowed to wear the Hants regtl badge.