Infinitude of Fibonacci gap fibers

For a positive integer $\ell$ and an integer $m$, let

$$D_\ell(m)=\{n:\exists k\geq 1\text{ such that }n=d_k,\ d_{k+\ell}-n=m\},$$

and define $U_\ell(m)$ analogously. Infinitude conjecture for gap fibers. For every $m\in D_\ell$,

$$|D_\ell(m)|=\infty.$$

The conjecture expresses the expectation that every admissible down-gap occurs infinitely often; the source motivates it by a stronger positive-density suspicion, but leaves the stated infinitude claim open.