Maximum entropy probability distribution with known quantile

For continuous distributions on x>0 with known mean m, the exponential distribution f(x) = (1/m)exp(-x/m) is the maximum entropy distribution, with entropy H(f) = ln(m)+1. I have a problem where I know the P-th quantile Q and I want to know the maximum entropy distribution with that quantile.

The exponential distribution with P-th quantile Q has mean m = Q/ln(1-P). As stated this is the maximum entropy distribution for all distributions with mean m. Is it also the maximum entropy distribution over all continuous distributions with P-th quantile Q? If not, what is the maximum entropy distribution. Any help greatly appreciated.

Thanks,

Trevor

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A related fact is that quantiles are not sufficient statistics for any distributions on $\mathbb{R}$, as noted here on page 17.