Unirationality conjecture for norm surfaces defined by sextic polynomials

Let $k$ be a number field and let $K/k$ be a cyclic extension of degree $3$. For a polynomial $f\in k[t]$ of degree $6$, let $\mathcal{S}_{f}$ denote the variety defined by the norm equation associated with $f$. A $k$-rational point on $\mathcal{S}_{f}$ is non-trivial if it is not one of the trivial points arising from the construction of the surface. Unirationality conjecture. If there exists a non-trivial $k$-rational point on $\mathcal{S}_{f}$, then $\mathcal{S}_{f}$ is $k$-unirational. The preceding approximation theorem establishes unirationality for suitably approximated sextics with a rational point at infinity, motivating this conjecture; the general assertion for sextic polynomials is left open in the source.