Two commercial payers for a short time, subscriber is same for both

Does anyone know what payer should be primary in the example below:

Client has commercial BCBS which termed 6/30/19 and subscriber/policy holder is husband. Same client gets new insurance HP effective 6/1/19 and husband is also policy holder for this insurance. 

This means for the month of June, there are 2 active payers, BCBS and HP. This likely happened due to a job change and the previous job with plan under BCBS let the coverage be active through the end of the month of June. New job with plan under HP started June 1.  

I'm thinking BCBS would be primary for June based on that fact that they have had that insurance longer (was effective before HP). I have not asked the client and assume that they would not know as most clients don't understand which plan should be primary or secondary. I've seen this dual coverage (with same policy holder) likely due to new jobs a few times and assume we need to bill BOTH payers while the 2 plans are active. It's just hard to know which should be primary.  Any thoughts or resources?  Thank you in advance.