Certifying determinantal representation conjecture for polynomials on ${\mathcal T}_{27}$

Let $p(w_1,\ldots,w_{27})$ be a polynomial of multidegree $(n_1,\ldots,n_n)$ satisfying, for some $\epsilon>0$, the bound

$$|p(w_1,\ldots,w_{27})|\geq \epsilon\left|r(\eta^{-1}(w))\right|^{-\sum_{j=1}^{27}n_j}$$

for $(w_1,\ldots,w_{27})\in\mathcal T_{27}$. Let $r$, $\eta$, and $\Omega(w)$ be the rational maps and matrix-valued map defined in the paper.

Certifying determinantal representation conjecture. There exists a polynomial $q(w_1,\ldots,w_{27})$ such that $pq$ has a certifying determinantal representation: there exist $k\in\mathbb N$ and an $m\times m$ matrix $A_0$, with $m\leq nk$ and ${\rm Im}\,A_0\geq0$, such that

$$p(w)q(w)=(w_1+i)^k\det\left(A_0+V^*(\Omega(w)\otimes I_k)V\right),$$

where $V$ is an $nk\times m$ matrix satisfying $V^*V=I_m$.

This conjecture seeks a determinantal certificate after multiplication by an auxiliary polynomial, under a quantitative lower bound on $p$ over the tube domain. The supplied context does not state whether it has been proved or disproved.