We continue the study of finite connected edge-bipartite graphs Δ, with m ≥ 2 vertices (a class of signed graphs), started in [SIAM J. Discrete Math. 27(2013), 827-854] and developed in [Fund. Inform. 139(2015), 249-275, 145(2016), 19-48] by means of the non-symmetric Gram matrix G ∨ Δ ∈ 𝕄 n ( ℤ ) defining Δ, its symmetric Gram matrix G Δ : = 1 2 [ G Δ ∨ + G Δ t r ∨ ] ∈ 𝕄 n ( 1 2 ℤ ) , and the Gram quadratic form q_Δ : ℤⁿ → ℤ. In the present paper we study connected positive Cox-regular edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense that the symmetric Gram matrix G_Δ ∈ 𝕄_n(ℤ) of Δ is positive definite. Our aim is to classify such Cox-regular edge-bipartite graphs with at least one loop by means of an inflation algorithm, up to the weak Gram ℤ-congruence Δ ~_ℤ Δ′, where Δ ~_ℤ Δ′ means that G_Δ′ = B^tr · G_Δ · B, for some B ∈ 𝕄_n(ℤ) such that det B = ±1. Our main result of the paper asserts that, given a positive connected Cox-regular edge-bipartite graph Δ with n ≥ 2 vertices and with at least one loop there exists a Cox-regular edge-bipartite Dynkin graph 𝒟_n ∨ {ℬ_n, 𝒞_n, ℱ₄, 𝒢₂} with loops and a suitably chosen sequence t • − of the inflation operators of one of the types Δ ′ ↦ t a − Δ ′ and Δ ′ ↦ t a b − Δ ′ such that the composite operator Δ ↦ t • − Δ reduces Δ to the bigraph 𝒟_n such that Δ ~_ℤ 𝒟_n and the bigraphs Δ, 𝒟_n have the same number of loops. The algorithm does not change loops and the number of vertices, and computes a matrix B ∈ 𝕄_n(ℤ), with det B = ±1, defining the weak Gram ℤ-congruence Δ ~_ℤ 𝒟_n, that is, satisfying the equation G_{𝒟_n} = B^tr · G_Δ · B.