Boavida-Weiss equivalence conjecture for the PL Stiefel space

Let $n-m\geq 3$. Write $\mathrm{V}^{pl}_{n,m}=\mathrm{PL}_n/\mathrm{PL}_{n,m}$ for the PL Stiefel space, and let $\mathrm{Operad}^h(E_m,E_n)$ be the derived mapping space of the little discs operads, pointed at the inclusion $E_m\subset E_n$. Boavida-Weiss equivalence conjecture. The natural map

$$\mathrm{V}^{pl}_{n,m}\to\mathrm{Operad}^h(E_m,E_n)$$

is an equivalence. This would identify the smoothing-theoretic and operadic deloopings of disc embedding spaces and would imply compatibility of the operadic delooping with the little discs action for $n\geq 5$.