Jonsson's partition-function conjecture for odd square-grid cylinders

Let $C_{m,n}$ be the square-grid cylinder, let $Z(C_{m,n})$ be the hard-particle partition function at activity $-1$, and suppose that $n$ is odd. Jonsson's cylinder partition-function conjecture. For all odd $n$,

$$Z(C_{m,n})= \begin{cases} -2 & \text{if }3\mid\gcd(m-1,n),\\ 1 & \text{otherwise.} \end{cases}$$

The conjecture is based on computations for $m\leq11$; the source does not state a resolution.