Coefficient log-concavity of the four-variable refinement

Prove that for every alternating link K, every exponent vector α∈Z^4, and every pair of indices i,j∈{1,2,3,4}, the coefficients of P_K satisfy c_α^2≥c_{α+e_i-e_j}c_{α-e_i+e_j}.

Background

For each exponent vector α, c_α denotes the coefficient of the corresponding monomial in P_K. The proposed inequality is the exchange-type log-concavity property satisfied by denormalized Lorentzian polynomials. The paper reports verification for alternating knots with at most eleven crossings but does not prove the inequality in general.

References

Second, $P$ appears to satisfy a log-concavity property.

Let $K$ be an alternating link, and given $\alpha \in Z4$ let $c_\alpha$ denote the coefficient of $x{\alpha_1}y{\alpha_2}z{\alpha_3}w{\alpha_4}$ in $P_K$. Then for any such $\alpha$ and any $i, j \in {1,2,3,4}$, we have $$ c2_\alpha \geq c_{\alpha + e_i - e_j} c_{\alpha - e_i + e_j}. $$

— An Alexander Polynomial Refinement for Alternating Links, with Trapezoidal Properties  (2608.28484 - Boninger, 28 Aug 2026) in Introduction, subsection “Fox's trapezoidal conjecture”; Conjecture 1.5 (labelled conjecture:lc_me)