Don't reverse! load_path cache

[Keno]

As noted in #50119 (comment).

[benlorenz]

Thanks for the quick fix, I think we were seeing the effects of this here: Nemocas/Nemo.jl#1497
During precompilation Pkg.project() sometimes stored the global project instead of the one where the package is active. I added proj = Pkg.status() to the global scope of Nemo and depending on how many packages are precompiled sometimes I got:

(jl_XcABI2) pkg> activate --temp
  Activating new project at `/tmp/jl_mTpSx2`

(jl_mTpSx2) pkg> add Nemo
... some precompilation ....

julia> using Nemo

julia> Nemo.proj
Pkg.API.ProjectInfo(nothing, nothing, nothing, false, Dict{String, Base.UUID}(), "/home/datastore/lorenz/software/julia/depot/environments/v1.10/Project.toml")