Chen's conjecture on the cohomological invariance of the twisted path threshold

Let $\chi>0$ be a closed positive $(1,1)$-form, and define $R(\chi)$ to be the supremum of times $t\in[0,1]$ for which the twisted continuity-path equation is solvable. Chen's conjecture. For any two closed positive $(1,1)$-forms $\chi_1$ and $\chi_2$ in the same cohomology class,

$$R(\chi_1)=R(\chi_2).$$

The source notes that the analogous assertion for the classical Aubin path was proved, but does not resolve the twisted statement.