A Conjecture for a Perrin-like sequence

Conjecture

Let $ W(k) $ be defined as:
$$
W(0) = 1, \quad W(1) = 0, \quad W(2) = 2,
$$
and for $ k > 2 $, recursively by:
$$
W(k) = 3W(k-2) + W(k-3).
$$

Then, for $ n > 0 $ such that $ n \equiv \pm 2, \pm 4 \pmod{9} $, $ n $ is prime if and only if all of the following conditions hold:

1. $$
   W(n) \equiv 0 \pmod{n},
   $$
2. $$
   W(n+1) \equiv -1 \pmod{n},
   $$
3. $$
   W(n+2) \equiv 1 \text{ or } W(n+2) \equiv -2 \pmod{n},
   $$
4. $$
   W(n^2+1) \equiv -1 \pmod{n},
   $$
5. $$
   W(n^2+2) \equiv 1 \text{ or } W(n^2+2) \equiv -2 \pmod{n}.
   $$