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We disprove a conjecture of Berthe et al.\\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897--904), which states that $\\chi_k'(G)\\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\\{k,2k\\}$ and a specified vertex partition. 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A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897--904), which states that $\\chi_k'(G)\\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\\{k,2k\\}$ and a specified vertex partition. 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