Let me sketch a couple of classical arguments that I think can be
argued to ask space time dimension to compactify to D5.

Consider a bound gravitational orbit around a mass m. Kepler third law
for generic dimension is T^2=R^(D-1). For D=4 this is the usual law.
And if instead of the radious of the orbit we consider the total area
of the orbit, we can also write it as T^2=A^((D-1)/2). Here we see
that something special happens at D=5: the period in this space time
depends linearly of the area.

If we consider in detail the area sweept by a particle as time goes,
we appreciate that D5 and D5 are different issues: The area goes

$$2 A(t) = G^1/2 m^1/2 R^((5-D)/2) t $$

Thus for D5, for a given time interval the area decreases with radius
of the orbit, while for D5, areas increase with the radius of the
orbit.
It should be a surprise if this change in the trend of gravitational
bound states were not noticed in the spectrum of any theory having
classical newtonian gravity as a limit. Thus D5 should be a natural
point for compatitification.

In favour of LQG/discrete area theories, it is worth to notice what
happens if we ask A(t) to be a integer multiple of Planck area when t
is Planck time. Except for D=5, this requisite fixes a quantification
of the radius, but only for D=4 the gravitational constant G cancels
out, leaving just h, c, and m. This is another signal of the
privileged status of our current space time, or at least of its
delicate relationship with the concepts of time and area.

Alejandro Rivero