But that isn't his actual Will. Is it supposed to be the one he made in 1633, in England? To see some accounts, Cornwallis obtained administration by fraud.
A lawsuit TNA https://discovery.nationalarchives.gov.uk/details/r/C5779638 Hawley v Cornwallis claims Jerome died intestate.
Short title: Hawley v Cornwallis.
Plaintiffs: James Hawley.
Defendants: Thomas Cornwallis, administrator [the estate and debts] of Jerome Hawley in Maryland, America.
Subject: estate of Jerome Hawley, brother of James, who died intestate in Maryland in 1638, after journeying to Virginia in 1636 on the ship the Black George of Portsmouth, using James's money to equip himself with goods to trade for tobacco.
Another suit - TNA https://discovery.nationalarchives.gov.uk/details/r/C5780328 mentions that Jerome's widow refused administration
Short title: Hawley v Cornwallis.
Plaintiffs: James Hawley.
Defendants: Thomas Cornwallis.
Subject: estate of Jerome Hawley, who died in Maryland, America, and whose widow refused administration; Cornwallis took on the administration in August 1638 as the greatest creditor, and paid off local debts, but his letters of administration did not cover debts in England: Cornwallis believes Jerome was defrauding James (mentions Gabriel Hawley and James Neale in Virginia).