Generalized exponent conjecture for restrictions of representations

Let $G$ be a real semisimple group and $H\subset G$ a real semisimple subgroup, with $K^H=H\cap K$ maximal compact in $H$ and $(\mathfrak a^H_0)^+\subset\mathfrak a_0^+$. Let $\pi\in\widehat G$, let $\tau$ be a $K$-type of $\pi$, and let $\xi$ be a $\tau$-exponent of $\pi$. Generalized exponent conjecture. There exist a representation $\sigma\in\widehat H$ weakly contained in the restriction of $\pi$ to $H$, a $K^H$-type $\tau^H\subset\tau|_{K^H}$ of $\sigma$, and a $\tau^H$-exponent $\nu$ of $\sigma$ such that

$${\rm Re}(\nu)=\max\left({\rm Re}(\xi)|_{\mathfrak a^H}-\rho|_{\mathfrak a^H}+\rho^H,0\right).$$

This is proposed as a generalization of the preceding rank-one restriction proposition; the source gives no proof or resolution.