Even-$K$ extension of rank and order properties for $K$-Fibonacci sequences

Let $K$ be even, and let $F_{K,n}$ be the $K$-Fibonacci sequence. For a modulus $m$, let $\pi_K(m)$ denote its Pisano period, let $\omega_K(m)$ denote the number of zeros in that period, and let $\alpha_K(m)$ denote the index of the first zero. Even-$K$ extension conjecture. For every modulus $m>3$, the following statements hold:

• $\omega_K(m)=4$ if and only if $\alpha_K(m)\equiv\pm1\pmod{4}$.
• If $\omega_K(m)=2$, then $4\mid\pi_K(m)$ and $2\mid\alpha_K(m)$.
• If $4\not\mid\pi_K(m)$, then $\omega_K(m)=1$.

These are proposed as portions of the odd-$K$ theorem that might extend to even $K$. The source later gives a counterexample to the converse of the third assertion, but that does not refute the three one-way assertions stated here.