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Assuming a homogeneous medium.

The electromagnetic continuity equation goes as

∇⋅J + ∂ρ/∂t = 0

since J = σE, ρ = ɛ∇⋅E, and assuming the time dependence exp(-iωt)

we have

σ∇⋅E - iωɛ∇⋅E = 0

(σ - iωɛ)∇⋅E = 0

So, σ - iωɛ = 0 or ∇⋅E = 0

Is it right? Thank you.