Don't pay yet, until you have submitted what is called an "informal challenge" to the PCN. You should base your representation on what you have told us, and also mention that you have a witness who can confirm the conversation with the CEO. Don't be confrontational, just firm. Once you get a response, please post it here, and you can decide whether to then cough-up, (the discount will be re-offered, see back of your PCN), or take them all the way to London Tribunals.
Next enforcement stage is the service of a Notice to Owner. Are you the owner, and if so, is your V5C up-to-date, because the NtO is sent to the name and address on the V5C ?