The determinantal-complexity lower-bound conjecture for the constructed family

Let $a>0$ be a sufficiently small constant, let $l$ be the nearest odd integer to $n^a$, choose disjoint odd-size subsets $C_1,\ldots,C_k\subseteq[n]$ with $k=\lfloor n^{1-a}\rfloor$, choose a random regular non-bipartite graph $G_n$ on $n$ nodes of degree $\sqrt n$, and let $Q$ be the face of $\operatorname{NPT}(\det(T_G))$ obtained by setting the odd-size constraints corresponding to these subsets to equalities. Define $f_n=\det(T_G)|_Q$.

Constructed-family determinantal lower-bound conjecture. If $a>0$ is small enough, then, with high probability, $f_n$ cannot be expressed as a symbolic determinant of size at most $n^{1+\delta}$, for a sufficiently small positive constant $\delta$.

The family is approximable by symbolic determinants of size $n$ and has a polynomial-size symbolic determinant representation, so the conjecture would provide a strong separation between its approximative and ordinary determinantal complexity. The source does not establish this lower bound.