Main conjecture on the recursive structure of fusible numbers

Let ${\mathcal{F}}$ be the set of fusible numbers. For $a\in{\mathcal{F}}$, let $s(a)$ be its successor, let $m(a)$ denote the relevant minimum gap, and write

$$[\overline{a},\overline{s(a)})=\bigcup_{n\in\mathbb{N}^+}\mathcal{I}_{a,n}$$

as before. Main conjecture. For every $a\in{\mathcal{F}}$ and every $n\in\mathbb{N}^+$, the fusible numbers in $\mathcal{I}_{a,n}$ can be written as $s^n(a)\sim c$ for some $c\in{\mathcal{F}}$. Equivalently, $\mathcal{I}_{a,n}\cap{\mathcal{F}}$ is a translated copy of

$$[\overline{a}-2^{1-n}m(a),\overline{a})\cap{\mathcal{F}}$$

scaled by a factor of $1/2$. This conjecture proposes a self-similar description of the fusible numbers in the intervals between a number and its successor, following the failure of the earlier formula for all fusible numbers; the source provides no proof or resolution.