Smooth transfer conjecture for relative endoscopy of unitary symmetric spaces

Fix a relative endoscopic datum $(\xi,\alpha,\beta)$, and let

$$\mathfrak g=\mathfrak u(W),\qquad \mathfrak h^{\alpha,\beta}=\mathfrak u(V_a\oplus V_\alpha)\oplus\mathfrak u(V_b\oplus V_\beta).$$

The endoscopic space $\mathfrak h^{\alpha,\beta}_1$ has a contraction map

$$r_{\alpha,\beta}:\mathfrak h^{\alpha,\beta}_1\to \mathcal{H}erm(V_a)\oplus\mathcal{H}erm(V_b).$$

For matching elements, define the relative transfer factor by

$$\Delta_{\mathrm{rel}}((x_a,x_b),x)=\Delta(r_{\alpha,\beta}(x_a,x_b),r(x)).$$

Smooth transfer conjecture. For any relative endoscopic datum $(\xi,\alpha,\beta)$ and any $f\in C_c^\infty(\mathfrak g_1)$, there exists $f_{\alpha,\beta}\in C_c^\infty(\mathfrak h^{\alpha,\beta}_1)$ such that $f$ and $f_{\alpha,\beta}$ match.

This asserts the existence of smooth transfer functions for relative endoscopy of unitary symmetric spaces. The supplied text does not state whether the assertion has been proved or disproved.