Partition-function bound for collapsing conformal field theories

Consider a family of unitary conformal field theories with fixed central charge $c$, parameterized by $\varepsilon\to 0$, and closed surfaces $\Sigma$ in the family described by long flat tubes of lengths $l_i$ and radii $R_i$, with $\lambda_{\min}$ the minimal eigenvalue of $L_0+\overline{L}_0$ and the sphere partition function normalized by $Z(S^2)=1$. Partition-function bound conjecture. For the family of closed surfaces $\Sigma$ as above,

$$Z(\Sigma)\exp\left(-\frac{c}{6}l_i\lambda_{\min}\right)\leq \operatorname{const},$$

where the constant does not depend on $\Sigma$. The bound is intended to control partition functions along a collapse or double-scaling limit; the source gives no resolution.