Morten O. Ravn and Harald Uhlig (2002)
This paper complements these insights using two different analytical
approaches. The first approach uses the time domain and focuses on
the ratio of the variance of the cyclical component to the
variance of the second difference of the trend component: this ratio
is often used for calculating the smoothing parameter. For a
particular benchmark stochastic process, it is shown that time
aggregation changes this ratio by the fourth power of the observation
frequency. The second approach uses the frequency domain and
investigates the transfer function of the HP filter, thereby obtaining
a general result. Again, a change-of-variable argument shows that one
should adjust the HP parameter with approximately the fourth power
of the frequency change. Both approaches therefore yield a value
of approximately $1600 / 4^4 = 6.25$ for annual data, which is close
to the value of 10 given by Baxter and King (1999).
Source: Notes On Adjusting the HP-Filter for the Frequency of Observations
Mueller's result is not obvious to me given this rule. By that rule of thumb, a monthly parameter given an quarterly parameter of 900,000 is $900,000 / 3^4 \approx 11,111$ (because there 3 months in a quarter). It also gives a yearly parameter of $900,000 / 12^4 \approx 43$ (12 months in a year).
The following quote seems to support Mueller.
On suboptimality of the Hodrick–Prescott filter at time series
endpoints Hodrick and Prescott (1997) proposed, on somewhat subjective
grounds, a value λ = 1600 for quarterly data. However, it is desirable
to adjust this value when observations of different frequencies are
subject to the filter. Backus and Kehoe (1992) suggested an adjustment
of the value by multiplying the standard value of 1600 with the square
of the frequency of observations relative to quarterly data. For
example, the relative frequency is 3 for monthly data and 1/4 for
annual data. Hence, the corresponding values of the smoothing
parameter is $\lambda$ = 100 and 14,400 for annual data and monthly data,
respectively. This suggestion has been also used in commercial
packages such as EVIEW. We shall use these values throughout the
paper. With regard to the choice of the smoothing parameter, it is
worth noting that, in research that has gone largely unnoticed in this
field, Akaike (1980), while further allowing a seasonal component in
the decomposition, proposed precisely the HP approach together with a
data-dependent Bayesian procedure for the choice of $\lambda$.
So just as $1,600 (quarterly) \rightarrow 3^2 \cdot 1,600 = 14,400 (monthly)$ so to would $100,000 (quarterly) \rightarrow 3^2 \cdot 100,000 = 900,000 (monthly)$
Misea, Kimb, and Newboldc (2005)
But the cited Backus and Kehoe (1992) result seems to be superseded by the Ravn and Uhlig (2002) result, so this may not reflect the state of the art thinking on this matter.