Relative category and monoidal topological complexity

If a map f has a homotopy retraction, then Doeraene and El Haouari conjectured that the sectional category and the relative category of f are the same. In this work we discuss this conjecture for some lower bounds of these invariants. In particular, when we consider the diagonal map, we obtain some results supporting Iwase–Sakaiʹs conjecture which asserts that the topological complexity is the monoidal topological complexity.