Balanced-border characterization of half-frequency words

Let $w$ be a binary word. A balanced border is a border of $w$ whose numbers of $0$s and $1$s are equal; the statistics $\rho^w$ and $q^w$ are the letter-frequency quantities associated with forbidding $w$. Balanced-border conjecture. The word $w$ satisfies

$$\rho^w=\frac{1}{2}$$

if and only if $w$ has balanced borders. The same equivalence holds with $\rho^w$ replaced by $q^w$. This conjecture would characterize exactly when either frequency statistic remains $1/2$ without requiring an evident $0\leftrightarrow1$ symmetry. The source says that a computer search suggests the claim and gives no proof or resolution.