I am working around with evaluation copies of JobCost, AR,AP & Invoice. I make separate sample source entries from Invoice and AP and post the retrieved transactions in JobCost. I find that the following information need to be addressed in Adagio JobCost. In my opinion the Job Status Report at transaction detail level should reflects the relevant data particularly in its reference and description fields

These are the data that were posted from sample entries

In the Billing/Payments Heading
Reference / Description
"Blank" / "Blank"
In the Costs Heading

16characters
truncated name
of a AP / "Blank"

My suggestion is
In the Billing/Payments Heading
ARInvoice No / AR Name

In the Costs
Heading
APInvoice No / AP Name

The AR & AP default GL integrations are set as follows:
Send to GL Reference field = Doc Number
Sent to GL description field = Customer/VendorName