An exact counterexample to odd-order Hankel H-eigenvalue inheritance

Take $m=3$, $q=2$, $n=2$ and the common generating vector $h=(2,0,1,0,2,0,-1)$, and let $A$ be the order-three, dimension-three Hankel tensor and $B$ the order-six, dimension-two Hankel tensor it generates. The first component of $Ax^2$ equals $(x_1+x_3)^2+x_1^2+x_2^2+x_3^2$, which is positive for every nonzero real $x$, so every real H-eigenvalue of $A$ is strictly positive. For $v=(0,1)$ one has $Bv^5=(h_5,h_6)=(0,-1)$, so $-1$ is an H-eigenvalue of $B$. Hence the inheritance of nonnegative H-eigenvalues from odd lower order fails. The premise is not vacuous: a root of $2t^4+2t^3+3t^2-4t-1$ in $(0,1)$ yields a real H-eigenpair of $A$. The example is consistent with the even-lower-order theorem and with inheritance from a positive-semidefinite associated Hankel matrix, a hypothesis that fails here because $h_6=-1$.