Full transfer conjecture for local Arthur parameters

Let $\Delta=((T_i,d_i,\eta_i,\lambda_i))_i$ be a shape and let $\varphi_v$ be a trace-positive test function on $G_v$. For every $\psi=\bigoplus_i\psi_i[d_i]$ in $\Delta$, with $\psi_i\in(T_i,1,\eta_i,\lambda_i)$, Full Transfer conjecture. There are trace-positive functions $\varphi_{i,v}$ on the appropriate quasi-split unitary groups of rank $T_i$ such that

$$\operatorname{tr}_\psi\varphi_v=\prod_i\operatorname{tr}_{\psi_i}\varphi_{i,v}.$$

The paper states this as a desired equality because exact transfer results at general places are unavailable; the conjecture is open.