Primitivity rank bound in terms of stable commutator length

Let $w\in\mathbb{F}$ be a word in a free group, let $\pi(w)$ be its primitivity rank, and let $\mathrm{scl}(w)$ be its stable commutator length. Primitivity-rank inequality. For any $w\in\mathbb{F}$,

$$\pi(w)\le 2\cdot\mathrm{scl}(w)+1.$$

In the source this is presented as equivalent to the universal character conjecture for polynomial irreducible characters of $U(N)$, using extremal-surface asymptotics. Its resolution is not stated.