Degree-diameter problem $N(5,7)$

The largest possible number of vertices in a graph of maximum degree $5$ and diameter $7$.

Lower bound: $5516$
Upper bound: Unknown

Updates

2008 Lower bound: $5516$
Loz, E., & Sirán, J. (2008). New record graphs in the degree-diameter problem. Australasian Journal of Combinatorics, 41, 63.
[via Table of the largest known graphs of a given diameter and maximal degree - Wikipedia]