The interlace-polynomial gap conjecture

Let $q_G(x)$ denote the interlace polynomial of a graph $G$. There are constants $c_1,c_2,\ldots$ with

$$c_1=\frac12>c_2>\cdots$$

such that, for every $k\geq 1$ and all sufficiently large $n$, there are graphs $G_1,\ldots,G_k$ of order $n$ satisfying

$$q(G_i)(1)=c_i2^n$$

for $1\leq i\leq k$, and every graph $G$ of order $n$ with $q_G(1)\geq c_k2^n$ satisfies $q_G(1)=c_i2^n$ for some $i\leq k$. Gap conjecture. The values of $q_G(1)$ above each fixed threshold eventually consist precisely of a discrete sequence of asymptotic levels. This conjecture concerns the unexplained gaps between successive large values of the interlace polynomial.