The Euler Sombor (\( EU \)) index of a graph \( G \) is defined as \[ EU(\mathit{G})=\sum \limits_{{\mathit{x}}{\mathit{y}}\in E(\mathit{G})}\sqrt{d_G^2(x)+d_G^2(y)+d_G(x)d_G(y)}, \] where \( d_G(x) \) and \( d_G(y) \) denote the degrees of vertex \( x \) and \( y \) in \( G \), respectively. Biswaranjan Khanra, Shibsankar Das [Euler Sombor index of trees, unicyclic and chemical graphs, MATCH Commun. Math. Comput. Chem. 94 (2025) 525-548], posed an open problem about determining the extremal values and extremal graphs for the Euler Sombor index among all connected graphs with a given diameter. In this paper, we solve this problem for maximum Euler Sombor index of unicyclic graphs with given diameter. Additionally, we propose a set of open problems for future research.