Log-convexity conjecture for admissible bundles

Let $\bar{M}$ be the curve under consideration, and let $\cal{L}$ and $\cal{M}$ be two bundles on $\bar{M}$ given by gluing conditions. Log-convexity conjecture. If both $\cal{M}\otimes\cal{L}$ and $\cal{M}\otimes\cal{L}^{-1}$ are admissible, then $\cal{M}$ is admissible as well. This is presented as a statement the authors were unable to prove; it expresses a log-convexity property of admissibility under tensoring by a bundle and its inverse.