The interval representation conjecture for fusible numbers

Given a fusible number $x\in\mathcal F$, let $y=s(x)$ be its successor, set $m=y-x$, and define

$$\ell_n=y+1-2^{1-n}m$$

for $n\in\mathbb N$, with

$$I_{x,n}=[\ell_n,\ell_{n+1}).$$

Also, write

$$s^{(n)}(x)=x+(2-2^{1-n})m.$$

Interval representation conjecture. Every fusible number in $I_{x,n}$, for $n\geq 1$, can be written as

$$s^{(n)}(x)\sim z$$

for some

$$z\in\mathcal F\cap [x+1-2^{1-n}m,x+1).$$

This conjecture proposes a uniform description of the fusible numbers in each interval between successive iterates of the successor operation. Its status is not established by the supplied source context.