Upper-bound conjecture for Δ-invariants of negative-definite plumbed manifolds

Let $Y$ be a negative-definite plumbed manifold, let $\spc{Y}$ denote its set of $spin^c$ structures, and let $\Delta_b$ be the corresponding invariant for $b\in \spc{Y}$. Let $\gamma(Y)$ denote the invariant appearing in the upper bound. Upper-bound conjecture.

$$\min_{b \in \spc{Y}} \Delta_b \leq -\frac{\gamma(Y)}{4}+\frac12.$$

The conjecture asserts that at least one $spin^c$ structure avoids the cancellations that can make individual upper bounds ineffective. Its status is not determined by the supplied source context.