on your honour
on (one's) honour
1. With utmost sincerity; with one's serious promise or oath. Primarily heard in UK, Australia. On my honour, I swear to you that I did not steal that money! If, on your honour, you promise to behave yourself, I'll let you go to the party with your friends.
2. With a sincere intent (to do something) without being watched or scrutinized; according to the honour system. Primarily heard in UK, Australia. To save on the cost of employing daytime staff to mind the till, we ask patrons on their honour to leave the correct payment for their items.
on your ˈhonour(old-fashioned)
1 used to promise very seriously that you will do something or that something is true: I swear on my honour that I knew nothing about this.
2 be trusted to do something: You’re on your honour not to go into my room.