In the outer solar system there is a region among the most distant Trans-Neptunian Objects (TNOs) where no objects have perihelia (closest approach to the sun) between roughly 50 au and 65 au. This apparent gap is statistically unlikely to be the result of observational biases as objects within this gap would be approximately five times brighter and easier to observe than objects beyond the gap such as Sedna. We have developed a suite of observational survey simulations in which we calculate the likelihood of several underlying object distributions fitting the observed extreme (high eccentricity, semi-major axis, and perihelion) TNO distribution. These simulations show that a bimodal Gaussian distribution in perihelion is a better fit (compared to a power-law distribution) to the observed distribution by several orders of magnitude, implying that the gap is a separation between two distinct outer solar system populations. We also ran hundreds of dynamical simulations to explore the effect a hypothetical distant giant planet would have on different underlying extreme TNO distributions. We find that simulations that do not contain a distant giant planet are unable to reproduce the apparent gap or the bimodal distribution of observed objects. Simulations including the hypothesised planet, however, can create the observed gap for some planet parameter combinations. The effectiveness of the planet at making the gap is dependent on the initial distribution of TNOs with Kuiper belt-like configurations producing the best results. Additionally, we find that extreme TNOs are scattered outward to high perihelia by the planet, hence, the gap may serve as a transition region between objects that are dominated by their interactions with Neptune and objects controlled by a distant giant planet.