No. Once it has reached your BAEC account it is yours. You earned it through spend, not by promising to remain a Chase customer.
No one on FlyerTalk has presented a situation where BA or Chase has actually enforced this requirement, but folks need to be aware of what the Travel Together Ticket Terms and Conditions actually say about this issue:
"Members must remain eligible for this promotion at the time of travel. BA and Chase reserve the right to withdraw any vouchers in the event that the Member is in breach of the terms of the BA/Chase Credit Card or the BA Executive Club Terms and Conditions."
Since the Terms and Conditions require you to "remain eligible for this promotion," it is significant to note that "eligibility" requires you to be a current cardholder:
"You are only eligible to receive a Travel Together Ticket (“Voucher”) for a companion if you are a member of the BA Executive Club and are the main account holder within a BA/Chase Credit Card account (a “Member”)."
You will need to judge for yourself whether there is a risk here.....