(This is essentially a continuation of my previous question, here.)

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. Further assume (though you can remove this assumption if you like) that each ball has positive finite volume. Let $M$ be the *uncentred* Hardy-Littlewood maximal operator, given by
$$Mf(x) := \sup_{B \ni x} \frac{1}{\mu(B)} \int_B |f(y)| \; d\mu(y)$$
with the supremum taken over balls containing $x$.

In my previous question, I asked when $M$ is not of weak type $(1,1)$, and was presented with an example ($\mathbb{R}^2$ with Gaussian measure) which isn't of weak type $(1,1)$, but which is of strong type $(p,p)$ for all $p>1$. (See Sjögren ('83), and Forzani, Scotto, Sjögren, Urbina ('02).)

Now I'm interested in the following question: are there any metric measure spaces $(X,d,\mu)$ for which $M$ is of strong type $(p,p)$ if and only if $p > p_0$ (or $p \geq p_0$) for some $p_0 > 1$ (note the strict inequality here)? In other words, do we ever have some *but not all* midpoint strong type estimates?

I know that there are maximal operators out there for which this is true, but I'm specifically interested in the uncentred Hardy-Littlewood maximal operator (and to some extent the centred version, though we have to assume some regularity of the measure here to ensure measurability).