Zhang's second conjecture on the lambda invariant of polarized metrized graphs

Let $f:X\to Y$ be a semistable fibration, let $CV(f)$ be the set of singular fibers, and let $\Gamma_y$ be the polarized metrized graph of the fiber over $y\in CV(f)$. For each node type $i$, let $\delta_i(\Gamma_y)$ denote its number. Let $\lambda(\Gamma)$ be Zhang's lambda invariant. Zhang's second conjecture. For every $y\in CV(f)$,

$$\lambda(\Gamma_y)\geq\frac{\bar{g}}{8\bar{g}+4}\delta_0(\Gamma_y)+\sum_{i\geq1}\frac{i(\bar{g}-i)}{2\bar{g}+1}\delta_i(\Gamma_y).$$

This inequality would yield a second proof of the stated slope inequality; the source gives no resolution status.