Jin's conjecture on almost internal logarithmic-differential pullbacks

Work in a sufficiently saturated differentially closed field $\mathcal{U}\models\mathrm{DCF}_0$ of characteristic zero, with derivation $\delta$, and let $F<\mathcal{U}$ be an algebraically closed differential subfield. Write $\mathcal{C}$ for the field of constants of $F$. Let $p\in S_1(F)$ be almost $\mathcal{C}$-internal, and let $q=\log_{\delta}^{-1}(p)$ be its pullback under the logarithmic derivative

$$\log_{\delta}(x)=\frac{\delta(x)}{x}.$$

Here $u\models q$ means that $\log_{\delta}(u)\models p$ and $u\notin\operatorname{acl}(F,\log_{\delta}(u))$.

Jin's conjecture. The following are equivalent:

1. $q$ is almost $\mathcal{C}$-internal.
2. The map $\log_{\delta}:q\to p$ is almost split.
3. There is an integer $k\neq 0$ such that, for some $u\models q$, there are $w_1,w_2$ satisfying

$$u u u^k=w_1w_2,$$

with $w_1\in\operatorname{dcl}(F,\log_{\delta}(u))$ and $\log_{\delta}(w_2)\in\operatorname{dcl}(F)$.

This conjecture characterizes almost $\mathcal{C}$-internality of the logarithmic-differential pullback through an almost-splitting condition and an explicit multiplicative decomposition. The source attributes the conjecture to Jin's thesis, but the supplied material gives no resolution status.