The barycentric isoperimetric inequality has been proved by B. Fuglede in the 90’s for convex sets and more recently by C. Bianchini, G. Croce and A. Henrot in the planar case for connected sets and by myself and A. Pratelli for bounded sets in any dimension. We will discuss a conjecture by C. Bianchini, G. Croce and A. Henrot on the set that optimizes the constant in the planar case among connected sets, providing an equivalent formulation.