I was using elnec in that example. But I was using "free space".
I was trying to leave the ground out of it...
Also those models were all elevated at 40 ft at the base.
At that height using real ground/average, it had about 3.8 dbi
at 8 degrees.
In comparison, the 1/4 GP with sloping radials did about 1.8 dbi
free space. Real ground at 40ft, was about 3.0 dbi at 9 degrees.
So about .8 db difference using real ground.

The main reason I was sort of skeptical about the 3.85 db
increase was due to your next to the last statement.
I didn't think the close spacing would give that much gain.
As I quick test, I just tried adding another element above
the original, in that model. I used 2 sources to feed.
With the very close spacing of the two elements, the free
space gain was 3.59 dbi...A tad less than I expected...
The real ground numbers were 5.53 dbi, at 7 degrees.
So almost a perfect 2 db increase in that model over
real ground. As a comparison, with about a 1/2 wave
spacing, I got 7.23 dbi using the same real ground.
About 3.4 db increase...Could probably get more
with a wider spacing...
It's this spacing thing that made me dubious...
I couldn't see 3.85 extra, being the elements were
very close using the stub.

BTW...all the models I had were for 10m, at 40 ft....
MK