Fixed points of the Jacobsthal sequence

Let the Jacobsthal sequence be defined by $J_0=0$, $J_1=1$, and $J_n=J_{n-1}+2J_{n-2}$, and let $m>1$ be an integer. A modulus $m$ is a fixed point when its Pisano period equals $m$. Fixed-point conjecture for the Jacobsthal sequence. $m$ is a fixed point if and only if

$$m=2\cdot3^k\qquad\text{for }k=1,2,\ldots.$$

The preceding discussion identifies the Jacobsthal sequence as a binary recurrence with $b=2$, contrasting it with the $K$-Fibonacci sequences studied in the paper. The source gives no resolution or supporting evidence beyond this stated claim.