Polynomial numerator conjecture for amplituhedron forms

Use coordinates $S,A,B$ for a flag $(W,U)\in{\rm Fl}(\ell,k+\ell;n)$, where $W$ is the row span of $[S\mid\operatorname{Id}_\ell\mid B]$ and $U$ is the row span of

$$\begin{pmatrix} S & \operatorname{Id}_\ell & B\\ \operatorname{Id}_k & 0 & A\end{pmatrix}.$$

Let

$$Z=\begin{pmatrix}\operatorname{Id}_k&-S&0\\0&-B&\operatorname{Id}_\ell\end{pmatrix}.$$

Polynomial numerator conjecture. There exists a polynomial $H(S,A,B)$ in the entries of $S,A,B$ such that

$$\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,m}(Z)}=\frac{H(S,A,B)\,d^{k\times m}(A)}{\Delta_{k+\ell}^{\operatorname{circ}}(U)},$$

and

$$\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,\geq m}}=\frac{H(S,A,B)\,d^{k\times m}(A)\wedge d^{\ell\times k}(S)\wedge d^{\ell\times m}(B)}{\Delta_{k+\ell}^{\operatorname{circ}}(U)\,\Delta_{\ell}^{\operatorname{circ}}(W)}.$$

Here $W=Z^\perp$ is the row span of $[S\mid\operatorname{Id}_\ell\mid B]$. These formulas seek polynomial numerators for the amplituhedron and universal forms; the source gives no resolution.