If $F$ is MHR (log-concave), can the monopoly reserve price exceed the mean?

Question

Let $V\sim F$ be some positive valued random variable such that $F$ has a monotone hazard rate (alternatively $F$ is log-concave). Let $p = \operatorname{argmax}q(1-F(q))$. We call $p$ the monopoly reserve price, when $F$ is MHR it's well known that $p$ is unique.

I'm interested the relationship between the monopoly price and the mean, specifically I want to know if $p \le \Bbb E[V]$, or else be given an MHR distribution such that $p > \Bbb E[V]$. The best bounds I've seen in the literature come from Lemma 3.4 of Giannakopoulos et al. (2021)¹ which shows that $$\frac{p}{\mu} \le \frac{e}{e-1} = 1.58$$ by loosely using a result of Barlow et al. (1964)².

For the canonical MHR distributions like exponential, uniform, etc. $p$ is either equal to or less than $\mu$ so I believe if the result fails is should take some sort of construction.

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_References

_{[1] Giannakopoulos, Y., Poças, D., Zhu, K. (2021). Optimal Pricing For MHR and $\lambda$-Regular Distributions. ACM Transactions on Economics and Computation. 9(1).}

_{[2] Barlow, R. E., Marshall, A. W. (1964). Bounds for distributions with monotone hazard rate, i. Ann. Math. Stat. 35(3):1234-1257.}