Support theorem conjecture for complementary-rank hyperbolic Radon transforms

Let $0\leq j<k\leq n-1$ satisfy $j+k=n-1$, and let $r>0$. Let $S(\Gamma_H(n,d))$ denote the space of rapidly decreasing smooth functions on the hyperbolic Grassmannian, with all derivatives rapidly decreasing. Support theorem conjecture. If $f\in S(\Gamma_H(n,d))$ and

$$(R_Hf)(\mathbf z)=0\quad\text{for all }\|\mathbf z\|>r,$$

then

$$f(\mathbf t)=0\quad\text{for all }\|\mathbf t\|>r.$$

An analogous result is stated to be unknown in the complementary-rank case with $j>0$, so this conjectural support-preservation property remains open.