Peano dimension of fundamental groups

We define the Peano dimension for groups arising as fundamental groups, which generalizes the classical definition of geometric dimension of finitely presented groups. We conjecture that the Peano dimension of the fundamental group of a aspherical Peano continuum $X$ is equal to the homotopy dimension of $X$. We prove the conjecture for one-dimensional or planar Peano continua. This answers a question posed by Cannon and Conner in 2007 concerning the homotopy dimension of planar sets.