In 1973, Neil Sloane published a very short paper introducing an intriguing problem: Pick a decimal integer n and multiply all its digits by each other. Repeat the process until a single digit ∆(n) is obtained. ∆(n) is called the multiplicative digital root of n or the target of n. The number of steps Ξ(n) needed to reach ∆(n), called the multiplicative persistence of n or the height of n is conjectured to always be at most 11. Like many other very simple to state number-theoretic conjectures, the multiplicative persistence mystery resisted numerous explanation attempts. This paper proves that the conjecture holds for all odd target values: – If ∆(n) ∈ {1, 3, 7, 9}, then Ξ(n) ≤ 1 – If ∆(n) = 5, then Ξ(n) ≤ 5 Naturally, we overview the difficulties currently preventing us from extending the approach to (nonzero) even targets.