The 5-connectivity conjecture for block and hole graphs

Let $\hat{G}$ be a block and hole graph with one pentagonal block and two quadrilateral holes, or, two quadrilateral blocks and one pentagonal hole. The 5-connectivity conjecture. If $\hat{G}$ is $5$-connected, then it is minimally $3$-rigid. This is one of the conjectures paraphrased from earlier work; the surrounding theorem establishes the corresponding single-block and single-hole cases, while the general situation addressed here is not resolved by the supplied text.