In discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $\mathbb Z^2$, and each finite upright path in $\mathbb Z^2$ is ascribed the weight given by the sum of values of its vertices. The weight of a collection of disjoint paths is the sum of its members' weights. The notion of a geodesic, a maximum weight path between two vertices, has a natural generalization concerning several disjoint paths: a $k$-geodesic watermelon in $[1,n]^2\cap\mathbb Z^2$ is a collection of $k$ disjoint paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have been largely unexplored beyond the $k=1$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $k=1$ the exponents that govern fluctuation in weight and transversal distance are $1/3$ and $2/3$; that is, typically, the weight of the geodesic on the route $(1,1) \to (n,n)$ fluctuates around a dominant linear growth of the form $μn$ by the order of $n^{1/3}$; and the maximum Euclidean distance of the geodesic from the diagonal has order $n^{2/3}$. Assuming a strong but local form of convexity and one-point moderate deviation bounds for the geodesic weight profile---which are available in all known exactly solvable models---we establish that, typically, the $k$-geodesic watermelon's weight falls below $μnk$ by order $k^{5/3}n^{1/3}$, and its transversal fluctuation is of order $k^{1/3}n^{2/3}$. Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes, which improve on estimates obtained via tools from the theory of determinantal point processes available in the integrable setting.