Recursive elimination-equation conjecture for three forms

Let $f$ and $g$ be the forms in the construction preceding the claim, and define

$$h_1=\det(f,g)_{(s,t)},\qquad h_{i+1}=\det(f,h_i)_{(s,t)}.$$

Let $L=(f,g,h_1,\ldots,h_n)$.

Recursive elimination conjecture. For arbitrary $n$,

$$L=(f,g,h_1,\ldots,h_n).$$

The source reports computational verification in degrees $5$ and $6$ and conjectures the displayed description for arbitrary $n$; its general status is unresolved in the source.