Equality of the Fibonacci gap sets

Let $D_\ell$ and $U_\ell$ be the down-gap and up-gap sets, respectively, defined by

$$D_\ell=\{d_{k+\ell}-d_k:k\geq 1\},$$

with $U_\ell$ defined analogously. Equality of the Fibonacci gap sets. For every positive integer $\ell$,

$$D_\ell=U_\ell.$$

The equality is known for $\ell=1$ and $\ell=2$, but the general case is posed as an open problem and is motivated by computations and the techniques used to establish results for small gap lengths.