Location of the minimal gap for twisted unknots

Prove that for the twisted unknot TU_n (obtained by applying n Reidemeister I twists), the smallest nonzero eigenvalue of the Khovanov Hodge Laplacian over all bigradings occurs in homological degree i=0 and quantum degree j=3−n.

Background

The authors paper spectral gaps of the Hodge Laplacian because these gaps control the efficiency of the quantum algorithm. For twisted unknots TU_n, extensive numerics up to n=10 suggest that the minimum spectral gap consistently occurs at bidegree (i=0, j=3−n). An analytic proof would clarify gap behavior for this family and support runtime estimates.