The GG conjecture on singularities after resolving the discriminant locus

Let $X$ be a projective complex variety of dimension $2n$, let $\mathcal L$ be a very ample invertible sheaf, and let $\zeta$ be a non-torsion primitive Hodge class. For an integer $k$, set $P=|\mathcal L^k|$. A resolution of the discriminant locus is a projective variety $S$ with a birational morphism $f:S\to P$ such that the inverse image of the discriminant locus is a divisor with normal crossings. If $\omega$ is a Deligne cohomology class mapping to $\zeta$, write $\nu(\omega,\mathcal L^k)$ for the associated normal function.

GG conjecture. For every non-torsion primitive Hodge class $\zeta$, there is an integer $k$ and a resolution $f:S\to P=|\mathcal L^k|$ of the discriminant locus such that, for any Deligne cohomology class $\omega$ mapping to $\zeta$, $f^*\nu(\omega,\mathcal L^k)$ is singular on $S$.

This strengthens the expectation that primitive Hodge classes give rise to singular normal functions by requiring singularity after a resolution of the discriminant locus and for every Deligne lift of the class. The supplied text gives no resolution status.