Conjecture on the instanton partition function for rational surfaces

Let $X$ be a rational surface, let $v_0$ denote the relevant charge, let $Z_{v_0}(\tau)$ be its partition function, let $\widetilde{Z}_{v_0}^{\int}(\tau)$ be the corresponding integrated generating function, let $\eta$ be the Dedekind eta function, and let $\chi(X)$ be the Euler characteristic of $X$. Rational-surface partition-function conjecture. Up to holomorphic anomaly,

$$Z_{v_0}(\tau)=\widetilde{Z}_{v_0}^{\int}(\tau)-\frac{1}{4\eta(2\tau)^{\chi(X)}}.$$

This conjecture predicts the correction relating the partition function to the integrated generating function for rational surfaces; the statement leaves the holomorphic-anomaly contribution unspecified.