Reference for classification of zero asymptotic variance in Litt’s game

Identify an existing published proof showing that, in Litt’s game on a binary alphabet with equal‑length words, the asymptotic variance o^2 equals zero only in the specific cases A = HT^{l−1} and B = T^{l−1}H (and symmetric variants), as formalized in Theorem 7.1.

Background

The authors prove that the asymptotic variance vanishes only for a particular family of degenerate binary words (and symmetries), noting that Basdevant et al. stated the condition without proof and that they are unaware of any published proof elsewhere.

A definitive reference would situate this classification within the literature and confirm whether the result was previously established.