  Our main objective in the present article is to study the factorization
problem for triple-product $p$-adic $L$-functions, particularly in the
scenarios when the defining properties of the $p$-adic $L$-functions involved
have no bearing on this problem, although Artin formalism would suggest such a
factorization. Our analysis is guided by the ETNC philosophy and it involves a
comparison of diagonal cycles, Beilinson--Flach elements, and Beilinson--Kato
elements, much in the spirit of the work of Gross (that is based on a
comparison of elliptic units and cyclotomic units) and Dasgupta (that dwells on
a comparison of Beilinson--Flach elements and cyclotomic units) for
smaller-rank motives.
