Multiplicativity conjecture for layered resultants

Let $f$, $g$, and $h$ be layered polynomials for which the layered resultants below are defined, and let $L$ be the layer structure; write $\mathrel{\underset{L}{\mid}}\joinrel\joinrel=$ for equality up to multiplication by a layer unit. Multiplicativity conjecture.

$$|\mathfrak R(f,gh)| \mathrel{\underset{L}{\mid}}\joinrel\joinrel= |\mathfrak R(f,g)|\,|\mathfrak R(f,h)|$$

and

$$|\mathfrak R(fg,h)| \mathrel{\underset{L}{\mid}}\joinrel\joinrel= |\mathfrak R(f,h)|\,|\mathfrak R(g,h)|.$$

The preceding results give bounds for permanents of layer Sylvester matrices in the primary case, but the multiplicative behavior of layered resultants asserted here remains unresolved in the supplied text.