Stable transfer conjecture for local representations

Let $\Delta=(\Delta_i)_i=(T_i,d_i,\eta_i,\lambda_i)_i$ be a shape of rank $N$, and let $\varphi_v$ be a test function on the twisted group $\widetilde{G}(N)$. Stable Transfer conjecture. There are test functions $\varphi_{i,v}$ on each $\widetilde{G}(T_id_i)$ such that, for every choice of $\psi_i\in\Delta_i$,

$$\operatorname{tr}_{\widetilde\pi(\psi)}\varphi_{i,v}=\prod_i\operatorname{tr}_{\widetilde\pi(\psi_i)}\varphi_{i,v}.$$

The conjecture is one of two local conjectures proposed to imply Full Transfer, apart from positivity. The supplied text gives no resolution status.