Divisibility of the Hurwitz form by the eigendiscriminant

Let $f$ be a cubic form in four variables, and let its eigenscheme determine a $3$-plane in the relevant Grassmannian. The eigenscheme is the scheme defined by the equations $\frac{\partial f}{\partial x_i}-\lambda x_i$ for $i=0,\ldots,3$; the eigendiscriminant is the homogeneous polynomial that vanishes when this eigenscheme is nonreduced or positive dimensional. The Hurwitz form is the polynomial on $\operatorname{Gr}(3,\mathbb{P}^{14})$ vanishing on $3$-planes that intersect the Veronese tangentially.

Divisibility conjecture. Restricted to the $3$-planes coming from eigenschemes, the eigendiscriminant divides the Hurwitz form.

The conjecture compares two discriminantal constructions associated with the Veronese embedding: the eigendiscriminant detects degeneracy of eigenschemes, while the Hurwitz form detects tangential intersection. The source presents this divisibility as an interesting comparison and does not provide evidence of a resolution.