Counterexamples to some duality principles of modular Nahm sums

Given any positive integer r r , Nahm’s problem is to determine all r × r r\times r rational positive definite matrix A A , r r -dimensional rational vector B B and rational scalar C C such that the rank r r Nahm sum associated with ( A , B , C ) (A,B,C) is modular. Around 2007, Zagier [ The dilogarithm function , Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 3–65] expected that if the rank r r Nahm sum for ( A , B , C ) (A,B,C) is modular, then so is the dual Nahm sum associated with ( A − 1 , A − 1 B , B T A − 1 B / 2 − r / 24 − C ) (A^{-1},A^{-1}B,B^\mathrm {T} A^{-1}B/2-{r}/{24}-C)