A symplectic capacity is a functor that to each symplectic manifold (possibly in a restricted subclass) assigns a nonnegative number. The Lagrangian capacity is an example of such an object. In this talk, I will state a conjecture concerning the Lagrangian capacity of a toric domain. Then, I will present two results concerning this conjecture. First, I will explain a proof of the conjecture in the case where the toric domain is convex and 4-dimensional. This proof makes use of the Gutt-Hutchings capacities as well as the McDuff-Siegel capacities. Second, I will explain a proof of the conjecture in full generality, but assuming the existence of a suitable virtual perturbation scheme which defines the curve counts of linearized contact homology. This second proof makes use of Siegel's higher symplectic capacities.