Generalized correction-term bounds and the diagonal intersection form conjecture

Let $Y$ be a rational homology sphere with $|H_1(Y;\mathbb Z)|=h$. Suppose that $Y$ bounds a negative-definite four-manifold $X$ whose first homology has no torsion. Write $d(Y,\mathfrak t)$ for the correction term associated to a $\text{Spin}^c$ structure, and let $\Delta_h$ denote the diagonal intersection form of rank $h$. Generalized correction-term bounds conjecture. One should have

$$\min_{\mathfrak t_0\in\operatorname{Spin}(Y)}d(Y,\mathfrak t_0)\geq \frac{1-h}{4},$$

and

$$\max_{\mathfrak t\in\operatorname{Spin}^c(Y)}d(Y,\mathfrak t)\geq \begin{cases}\left(1-\frac{1}{h}\right)/4 & \text{if $h$ is odd},\\[5mm] 1/4 & \text{if $h$ is even.}\end{cases}$$

If equality holds in either inequality, then the intersection form of $X$ is $\Delta_h$. The statement is presented as a consequence of Conjectures and; the supplied text gives no resolution, so its status remains open.