Periodization conjecture for the generalized sine-kernel Fredholm determinant

Let $I+V$ be the generalized sine-kernel operator (GSK), let $x$ be the large asymptotic parameter, and let $\nu$ denote the parameter whose integer shifts restore the determinant's periodicity. Write

$$\mathcal{A}[\nu](x)\sim \operatorname{det}[I+V]^{(0)}[\nu]\left(1+\frac{C_1(\log x)[\nu]}{x}+\dots+\frac{C_M(\log x)[\nu]}{x^M}+\dots\right),$$

for the formal asymptotic series corresponding to the non-oscillating part of the asymptotic series for $\log\operatorname{det}[I+V]$, where $C_k(X)[\nu]$ is a polynomial of degree $k$ in $X$ with coefficients that are functionals of $\nu$, and where each $C_k$ contains no oscillating exponentials of the form $\mathrm{e}^{\pm i x p_{\pm}}$. Periodization conjecture. The formal asymptotic series for the Fredholm determinant is

$$\operatorname{det}[I+V]\sim\sum_{n\in\mathbb{Z}}\mathcal{A}[\nu+n](x).$$

Equivalently, the asymptotic expansion restores the periodicity $\nu\to\nu+n$ for $n\in\mathbb{Z}$, and all oscillating terms can be deduced from the non-oscillating ones by integer shifts of $\nu$. This conjecture describes the expected structure of the full asymptotic series beyond the first terms, whose shifted form is exhibited by the preceding correction formula; its general validity is not established in the supplied text.