Hitchin WKB local-estimate conjecture

Let $\gamma:[0,1]\to\widetilde X^{\ast}$ be a noncritical path, meaning that the pullbacks $\gamma^{\ast}\operatorname{Re}\phi_i$ are distinct for every $t\in[0,1]$, and order them so that

$$\gamma^{\ast}\operatorname{Re}\phi_1>\gamma^{\ast}\operatorname{Re}\phi_2>\cdots>\gamma^{\ast}\operatorname{Re}\phi_r.$$

Let $h_t$ denote the family of harmonic maps in the Hitchin WKB problem and let $\overrightarrow d$ be vector distance. Hitchin WKB local-estimate conjecture. The same asymptotic estimate as in the classical WKB theorem should hold for the Hitchin WKB problem:

$$\frac{1}{t}\overrightarrow d\bigl(h_t(\gamma(0)),h_t(\gamma(1))\bigr)\sim(\alpha_1,\ldots,\alpha_r),$$

where

$$\alpha_i=\int_0^1\gamma^{\ast}\operatorname{Re}\phi_i.$$

The preceding theorem establishes this estimate for the complex or Riemann--Hilbert WKB problem; the Hitchin version is stated as an expected extension and is not resolved in the source.