Ba(OH)2 + H2SO4 Conductivity Problem?

This claim is partially right, but not completely right.

Ba(OH)₂ completely dissociates in aqueous solution to give Ba²⁺(aq) and OH⁻ ions, while H₂SO₄ completely dissociates in aqueous solution to give H⁺(aq) and SO₄²⁻(aq) ions. When the two solutions are mixed, Ba²⁺(aq) and SO₄²⁻(aq) ions combine to form BaSO₄ precipitate, and H⁺(aq) and OH⁻(aq) ions undergo neutralization to form water.
Ba²⁺(aq) + SO₄²⁻(aq) → BaSO₄(aq)
H⁺(aq) + OH⁻(aq) → H₂O(l)

Referring to the above two ionic reaction, the concentration of mobile ions in the solution decreases, leading to the electric conductivity also decreases. When both of the Ba(OH)₂ and H₂SO₄ completely react, the electricity conductivity of the mixed solution drops to zero.

no, most sulfonated salts in a aqueous solution remain in an ionic state that would make current possible.