Gan–Gross–Prasad local uniqueness conjecture for unitary groups

Let $E/F$ be a quadratic extension of local fields of characteristic zero, let $W$ be an $n$-dimensional hermitian space over $E$, and let $V=W\oplus^\perp e$ with $(e,e)=1$. Set $H=U(W)$ and $G=U(W)\times U(V)$, with $H$ diagonally embedded in $G$. Let $\varphi$ be a generic Langlands parameter for $G$, and let $\Pi^G(\varphi)$ be its $L$-packet. An irreducible representation is $H$-distinguished when its space of $H(F)$-invariant continuous linear forms is nonzero. Gan–Gross–Prasad conjecture. The $L$-packet $\Pi^G(\varphi)$ contains at most one $H$-distinguished representation.

This is the local uniqueness assertion in the Gan–Gross–Prasad conjecture for unitary groups. The source attributes it to Gan, Gross and Prasad and gives no resolution in the supplied text.