Joseph Morse, of Watertown - Disputed origins

Evidence needed to support Joseph Morse, of Watertown as son of Joseph Morse, of Ipswich & Dorothy Morse

Disputed Origins

Joseph's origins are unknown. He arrived in New England 1634 on the Elizabeth of Ipswich. This does not mean that he was from Ipswich only that he sailed from there.[1] Anderson comments that G. Andrews Moriarty in his pedigree of the Morse Family of Suffolk and Essex in England, placed this Joseph as son of Joseph (died 1637). He left a will naming sons Joseph and John, but Anderson says there is no evidence, beyond the will, that places Joseph of Watertown as son of Joseph of Ipswich.

https://www.wikitree.com/wiki/Morse-360

Re: Joseph Morse Sr. of Ipswich (1576-1646)

According to The Great Migration (Vo. V, Pages 169-170), this man immigrated to New England in 1637 (not 1635), and although this man's will mentions sons John and Joseph, there is no evidence that places Joseph Morse, of Watertown as a son of this man. The man who settled in Watertown gave his age as 24 in 1634, indicating he was born in about 1610 -- 3 years before the 1613 birth date that apparently has been associated for this man's son John.

Is there new scholarship (since 2007) that confirms this man as the father of Joseph Morse of Watertown? Or that places his immigration in 1634?

From The Great Migration: