There was never a connection. And there is no such letter from
Diggs to Phi beta sigma asking them to join Kappa Alpha Psi. No
sigma has seen such a letter nor members of Kappa Alpha Psi. There
were two letters written one from Diggs to college men that were
not organized explaining the orgins of Kappa, and another from a
fraternity to Kappa asking to join, but nothing came of it.

Yet, testimony to such a letter has persisted through the years
and it appears evident through the annals of history that some form
of communication did materialize between the fraternities of Kappa
Alpha Psi and the Phi Beta Sigma. According to page 97 of a book
entitied Black Greek-letter Organization in the Twenty-first
Century the following historical account is noted: “It was at this
time that the fraternity [Phi Beta Sigma] had another unique
opportunity to expand. In a letter dated December 11, 1915, Kappa
Alpha Psi Fraternity founder and Grand Polemarch Elder W. Diggs
offered to merge Kappa Alpha Psi with Phi Beta Sigma. At the time,
Kappa was establishing itself in the Midwest, while Sigma was
expanding in the East and South. The fraternity’s general board
considered the proposal, but in a reply dated December 18, 1915,
[Abram Langston] Taylor turned down the offer.”