I was reading the following question: A stochastic process that is 1st and 2nd order (strictly) stationary, but not 3rd order stationary

The following matrix is given representing the correlation structure of a stochastic process $X[t], X[t+1], X[t+2]$:

$$\left[\begin{array}{cc} \sigma^2 & a & b \newline a & \sigma^2 & c \newline b & c& \sigma^2 \end{array}\right]$$

I noticed that the correlation structure matrix is not a Toeplitz matrix. However, this seems to result in a contradiction: if you take $t=k$ then you find that the correlation between $X(k+1)$ and $X(k+2)$ is $c$, whereas if you take $t=k+1$, then you find that the correlation between $X(k+1)$ and $X(k+2)$ is $a$. This seems to mean that the matrix is not a valid correlation structure unless it is a Toeplitz matrix.

Is my reasoning valid?