0% vAT in subcontracting?

I made upholstered cabinet doors for a furniture maker who, in turn, received the order from a customer. Is this subcontracting?

The furniture maker says that I am not allowed to charge vAT on this invoice because it is an invoice between self-employed persons (the furniture is built-in, but the house is a new build, so it is not older than 10 years, which means that a reduced rate of 6% does not apply here).

Is this a case of ‘reverse charge vAT’?