Minimum distinct abelian squares in circular binary words conjecture

For a circular binary word of length $n$, let $MDC(n)$ denote the minimum number of distinct abelian squares it contains. The circular minimum conjecture. The minimum number is

$$\frac{n-1}{2}$$

if $n$ is odd, and this bound is attained only by $a^n$, $a^{n-1}b$, and their complements and conjugates; if $n$ is even, the minimum is

$$\frac{n-2}{2},$$

and this bound is attained only by $a^kb^{n-k}$ and its complement and conjugates, where $k\in\{1,3,5,\dots,n-1\}$. The claim is presented as suggested by computer experiments, and the source gives no proof or resolution.