Despite being composed exclusively of pentagonal and hexagonal faces, fullerene graphs (planar, cubic, and 3-connected) exhibit surprising properties. One of the most studied characteristics is the graph diameter, which remains difficult to determine in general. In 2012, Andova and Škrekovski formulated a conjecture regarding this parameter. They proposed that for a fullerene graph with \( n \) vertices, the diameter satisfies \( \mathrm{diam}(G) \geq \lfloor (5n/3)^{1/2} \rfloor - 1. \)

This conjecture is inspired by the study of a family of spherical fullerene graphs \( G_{i,j} \), with \( i, j \in \mathbb{N}^* \) and \( i \le j \), which also possess icosahedral symmetry. However, in 2023, Silva et al. proved the existence of infinite families of fullerene graphs with icosahedral symmetry, say \( G_{i,2i} \), that contradict this conjecture.

Thus, a natural question that arises is whether every graph \( G_{i,j} \), when \( j \) is a multiple of \( i \), satisfies the conjecture. In this paper, we provide a negative answer to this question and develop new techniques for determining the diameter of this family of graphs. These results completely settle this conjecture for this entire family of graphs.