Parity-dependent asymptotic domination of contractible Hamiltonian cycles

Let $h_m^c(n)$ and $h_m^{nc}(n)$ denote, respectively, the numbers of contractible and non-contractible Hamiltonian cycles in the grid cylinder graph $P_{m+1}\times C_n$, and let $\theta_{m,c}$ and $\theta_{m,nc}$ be the corresponding positive dominant characteristic roots. Parity-dependent domination conjecture. For fixed $m$ and $n\to\infty$,

h_m(n)=h_m^c(n)+h_m^{nc}(n)\sim\left\{\begin{array}{ll}a_{m,c}n\theta_{m,c}^n&\operatorname{if} $m$ is odd},\theta_{m,nc}^n&\operatorname{if} $m$ is even}. \end{array}\right.

This predicts that contractible Hamiltonian cycles asymptotically dominate when $m$ is odd, whereas non-contractible Hamiltonian cycles dominate when $m$ is even. The claim is motivated by computational data for $m\leq9$ and is presented as a conjecture for all $m$.