On denominators of consecutive $\mathrm{SL}(2,\mathbb{N})$-saturated Farey fractions

The sequence $({\mathcal{S}}_Q{)}_Q$ of $\mathrm{SL}(2,\mathbb{N})$-saturated Farey fractions was defined in our previous work by ${\mathcal{S}}_Q:=\{a\mathrm{/}q\in \mathbb{Q}\cap (0,1]:q+a+\bar{a}\le Q\}$, where $\bar{a}$ is the multiplicative inverse of $a(\mathrm{m}\mathrm{o}\mathrm{d}q)$ in $[1,q)$. Here, we prove that the set of $Q$-scaled denominators of consecutive fractions in ${\mathcal{S}}_Q$ is dense in the region $\mathcal{V}:=\{(x,y)\in [0,1{]}^2:\max \{(1-3x)\mathrm{/}2,2x-1\}\le y\le \max \{x,1-x\}\}$, and provide a formula for their distribution in $\mathcal{V}$ as $Q\to \mathrm{\infty }$.