Stabilization conjecture for the arc operad

Let $\mathcal{A}rc$ be the arc operad, and let ${}^{\rm Fat}{\mathcal S}t{\mathcal{GT}ree}$ denote the stabilized fat graph operad. A suitable stabilization conjecture for the arc operad. There is a suitable stabilization ${}^{\rm Fat}{\mathcal S}t\mathcal{A}rc$ of a thickening of $\mathcal{A}rc$ whose $0$--term is the disc and which contains ${}^{\rm Fat}{\mathcal S}t{\mathcal{GT}ree}$ as a suboperad; consequently, the group completion of

$$\amalg_{n\geq 0}{}^{\rm Fat}{\mathcal S}t\mathcal{A}rc(n)$$

is an infinite loop space. This would extend the infinite-loop-space construction from stabilized fat graphs to a stabilization of the arc operad; the source presents this as a suitable stabilization to be constructed, and no resolution is given.