EQ-type equals VEL-type for plane triangulation graphs

Let $4\mathcal{T}4$ be a plane triangulation graph, with EQ-type, VEL-type, and CP-type denoting the discrete types associated respectively with equilateral triangulation geometry, vertex extremal length, and circle packing. The EQ-type conjecture. For any plane triangulation graph, EQ-type coincides with VEL-type, and therefore with CP-type. For bounded-degree plane triangulation graphs, EQ-type and CP-type already coincide by quasiconformal comparison; the conjecture concerns the unbounded-degree case, where the relationship with VEL-type remains open.