The disjoint zero-block conjecture for the floor-function differences

Let $n\geq 1$ be an odd integer, and define

$$d_n(\ell):=\left\lfloor\sqrt{2\ell n}\right\rfloor-\left\lfloor\sqrt{(2\ell-1)n}\right\rfloor.$$

Suppose that $\delta:=d_n(\lambda)\geq 2$ for some $\lambda\geq 1$. Disjoint zero-block conjecture. Each such $\lambda$ corresponds to $\delta-1$ consecutive integers $\ell_{\lambda},\ldots,\ell_{\lambda}+\delta-2$ such that

$$d_n(\ell_{\lambda}+j)=0\qquad (j=0,\ldots,\delta-2),$$

and the sets of these zero values are disjoint. The statement describes the observed relationship between values of the floor-function differences: a value at least $2$ is paired with a consecutive block of zero values, with different blocks disjoint. The supplied text does not state whether this claim has been proved or remains open.