The interior problem conjecture for the Radon transform

Let $D$ be an open, bounded, convex domain in the plane and let $K$ be a closed subset of $D$. For $f\in C_c^{\infty}(\overline D)$, write

$$\mathcal R f(L)=\int_L f\,dx$$

for its Radon transform on a line $L$.

Interior problem conjecture. There exists a function $f\in C_c^{\infty}(\overline D)$, not identically zero in $K$, such that $\mathcal R f(L)$ vanishes for every line $L$ that intersects $K$.

The assertion is known for disks and ellipses by the Abel-integral argument and affine invariance. The supplied text says that it is not known for arbitrary bounded convex domains.