Isotopy conjecture for analytic continuation of spherical principal series

Let $G=\mathrm{SL}(n,\mathbb R)$, let $\mathrm{Fl}(\mathbb R^n)$ and $\mathrm{Fl}(\mathbb C^n)$ denote the real and complex complete flag varieties, and let $\mathcal L\subset\mathrm{Fl}(\mathbb C^n)$ be the union of flags containing a subspace on which the symmetric bilinear form $B(x,y)=\sum x_jy_j$ is degenerate. Let $\Delta$ be the discriminant submanifold in $\mathrm{GL}(n,\mathbb C)$, and let $g(s)$ denote the associated path of transformations. For a path $\gamma(t)$ in $\mathrm{GL}(n,\mathbb C)$ avoiding $\Delta$, with $\gamma(0)\in\mathrm{SL}(n,\mathbb R)$, the conjecture asserts the following. Isotopy conjecture. There is an isotopy $C(t)$ of the cycle $\mathrm{Fl}(\mathbb R^n)$ in $\mathrm{Fl}(\mathbb C^n)$ avoiding the submanifolds $\mathcal L$ and $\mathcal L\cdot g(s)$. This isotopy is intended to support analytic continuation of the $K$-finite functions occurring in the spherical principal series; the supplied passage does not state whether the conjecture has been proved or disproved.