Today Gathering For Gardner posted a video by Yoshiyuki Kotani called "Liar/Truth Teller Patterns on Square Planes".
The idea is that you fill a grid with knights and knaves so that both the knights and knaves can say "I am horizontally/vertically adjacent to $k$ knaves"—naturally the knights are telling the truth, and the knaves are lying.
Examples
For all of these examples, let $k=1$ and let . represent a knight and x represent a knave.
For $k=1$, the "boring" solutions are those that consist of horizontal or vertical strips, such as
x x . . x x .
x x . . x x .
x x . . x x .
x x . . x x .
x x . . x x .
x x . . x x .
x x . . x x .
Here is a non-boring solution on a $4 \times 4$ grid.
. x . .
. . . x
x . . .
. . x .
Similarly the video claims that there is exactly one non-boring solution for a $5 \times 5$ grid up to symmetry of the square:
. . x . .
x . . . x
. . . . .
. x x x .
. x x x .
Question
The video claims that there may be no solution for $k=1$ on the $6 \times 6$ board. That is, all of the solutions consist of horizontal/vertical strips. Is this the case? Is there a way to prove it?
Also, are there any corresponding OEIS sequences?