Zhang's first conjecture on the phi invariant of polarized metrized graphs

Let $f:X\to Y$ be as above, let $CV(f)$ denote the set of singular fibers, and for $y\in CV(f)$ let $\Gamma_y$ be the polarized metrized graph of the fiber. Write $\delta_i(\Gamma_y)$ for the number of nodes of type $i$, where type $0$ means that partial normalization at the node is connected and type $i\geq1$ is determined by the minimum arithmetic genus of the two components. Zhang's first conjecture. There is a positive continuous function $c(\bar{g})$ of $\bar{g}\geq2$ such that, for every $y\in CV(f)$,

$$\varphi(\Gamma_y)\geq c(\bar{g})\delta_0(\Gamma_y)+\sum_{i\geq1}\frac{2i(\bar{g}-i)}{\bar{g}}\delta_i(\Gamma_y).$$

This inequality would imply the effective Bogomolov conjecture and hence the Bogomolov conjecture; the source gives no resolution status.