In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of preservation):
Here’s some blue-sky numerology (I think Chris Godsil told me this originally, but I can’t remember).
Spectral theory tells us that an independent set in this hypothetical Moore graph can have at most 400 vertices, leaving 2850 left over.
These are magic numbers though, because 400 is the number of points in PG(3,7) and 2850 is the number of lines in PG(3,7). So perhaps we can construct a Moore graph with an independent set of size 400 by using the points of PG(3,7) as the independent set and the lines of PG(3,7) as the remaining vertices. The natural incidence between points and lines in PG(3,7) yields exactly the required number of edges between the two parts.
So this leaves us just the challenge of deciding adjacency among the 2850 “line-type” vertices.
If we take one of the 400 “point-type” vertices, then it has 57 line-type vertices as neighbours, and each of those has a further 49 line-type vertices as its neighbours. All 57.50 = 2850 of these must be distinct and so this configuration is a collection of 57 spreads of PG(3,7) that collectively use all the lines, or in other words a packing of PG(3,7).
So all we need to do is find 400 packings of PG(3,7) that fit together properly!!
My question is: have there been any results from (or even serious attempts at) taking this approach? Do we even know if this approach is compatible with the known properties that such a Moore graph must posess, if it were to exist (e.g.: automorphism group must have order $\leq$ 375 )?
 Macaj, Martin, and Jozef Širán. "Search for properties of the missing Moore graph." Linear Algebra and its Applications 432 (2010): 2381-2398.