The interval conjecture for stable commutator length in a rank-two free group

Let $F_2$ be the free group of rank $2$, and let $B_1^H(F_2)$ denote the space of real homologically trivial $1$-boundaries in $F_2$. An integral chain is an integral element of this space. The stable commutator length of such a chain is denoted by $\operatorname{scl}$.

Interval conjecture. The set of values

$$\{\operatorname{scl}(\Gamma):\Gamma\text{ is an integral chain in }B_1^H(F_2)\}$$

contains every rational number in the interval $[3/4,1]$.

The conjecture is motivated by a histogram of stable commutator lengths obtained from 7500 random realizations associated to a four-punctured sphere. It predicts that all rational values in the indicated interval occur, but the supplied text gives no resolution.