Continuity of the top Lyapunov exponent under a single uniform holonomy

Let $\hat{A}_k$ and $\hat{A}$ be continuous linear cocycles over the same hyperbolic system. Suppose that each cocycle is equipped with a uniform unstable holonomy $H^{u,k}$ and $H^u$, respectively, or with a uniform stable holonomy $H^{s,k}$ and $H^s$, respectively. For a cocycle with top Lyapunov exponent, write this exponent as $\lambda_{+}(\hat{A})$. Single-holonomy continuity conjecture. If

$$(\hat{A}_k,H^{u,k})\stackrel{C^0}{\longrightarrow}(\hat{A},H^u)$$

or

$$(\hat{A}_k,H^{s,k})\stackrel{C^0}{\longrightarrow}(\hat{A},H^s),$$

then

$$\lambda_{+}(\hat{A}_k)\to\lambda_{+}(\hat{A}).$$

This would relax the known continuity theorem requiring both uniform stable and uniform unstable holonomies to the existence of only one such holonomy. Results for cocycles over uniformly expanding maps, where the natural extension provides a uniform stable holonomy, motivate the conjecture; its general validity is not established in the supplied text.