Minimum semidefinite rank conjecture for graphs of large girth

Let $G$ be a connected graph on $n\geq 2$ vertices, and let $k\geq 4$ be an integer. The minimum semidefinite rank ${\rm msr}(G)$ is the smallest rank among all positive semidefinite Hermitian matrices with graph $G$. The girth of $G$ is the length of its shortest cycle.

Large-girth minimum semidefinite rank conjecture. If $G$ has girth at least $k$, then

$${\rm msr}(G)\geq \left(\frac{k-2}{k}\right)n.$$

This conjecture seeks to generalize the known lower bound ${\rm msr}(G)\geq n/2$ for connected triangle-free graphs, viewing triangle-freeness as a girth lower bound. Its status is not determined by the supplied source information.