In 2019, researches in Japan (M. Takamoto, H. Katori et al.) carried out experiments comparing two transportable ${}^{87}\text{Sr}$ clocks, one ("clock $\mathsf 1$") placed at the ground floor and the other ("clock $\mathsf 2$") placed in the observatory floor of the Tokyo Skytree broadcasting tower, nominally $450~{\rm m}$ above the ground floor. There's an article filed, with a public supplement and with a public article copy available.
The article introduces the symbols $\nu_{\mathsf 1}$ and $\nu_{\mathsf 2}$ to denote "the clock frequency at location $\mathsf 1$" and "the clock frequency at location $\mathsf 2$", respectively; and also referred to as "the clock laser frequencies".
Ramsey spectra (Excitation probability over Detuning) measured in the observatory floor and the ground floor with fit curves are shown (Fig. 2 b,c) with the maximum peak for clock $\mathsf 2$ at the reported Detuning value $\approx 21.18~{\rm Hz}$ and the maximum peak for clock $\mathsf 1$ apparently at Detuning value $0~{\rm Hz}$. (For reference, let's denote these two quantities as $f^{RSmax}_{\mathsf 2 \leftarrow\mathsf 1}$ and $f^{RSmax}_{\mathsf 1 \leftarrow\mathsf 1}$, resp.)
This is surely a quite significant finding, considering that
the full width at half maximum of the maximum peak of either Ramsey spectrum is apparently also in the order of $\approx 20~{\rm Hz}$,
the CIPM recommended frequency of the relevant unperturbed ${}^1 S_0 - {}^3 P_0$ optical transition of ${}^{87}\text{Sr}$ is given with accuracy better than $1~{\rm Hz}$, as $f({}^{87}\text{Sr}) = 429 \, \, 228 \, \, 004 \, \, 229 \, \, 873.7 (0.5)~{\rm Hz}$, and
the experiments involve corrections of about $+2.6~{\rm Hz}$, and systematic uncertainties much less than $\pm 0.1~{\rm Hz}$; as listed in Table S1 of the supplement.
Now, from the outset, in its abstract, the article claims that "A clock at a higher altitude ticks faster than one at a lower altitude, in accordance with Einstein’s theory of general relativity";
and it goes on to refer to "frequency shift $\Delta \nu = \nu_{\mathsf 2} - \nu_{\mathsf 1} \approx 21.18~{\rm Hz}$".
Therefore
My question:
Is it correct to conclude that clock $\mathsf 2$, while located at the observatory floor of the Tokyo Skytree broadcasting tower, ticked significantly faster than clock $\mathsf 1$, while located at the ground floor; in particular that (rounding down to integer ${\rm Hz}$)
the clock frequency at location $\mathsf 2$ had the value $\nu_{\mathsf 2} \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 897~{\rm Hz}$ while
the clock frequency at location $\mathsf 1$ had the value $\nu_{\mathsf 1} \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$ ?
(Or, to consider at least one alternative explicitly:
Is it instead correct to conclude that both clocks ticked as good as equally fast, both, at their respective locations, at approximately $\nu_2 \approx \nu_1 \approx 429 \, \, 228 \, \, 004 \, \, 229 \, \, 876~{\rm Hz}$; but the reported "frequency shift" value is instead attributable to the (suitably signed) difference between the tick frequency of a clock and the frequency of a receiver in response to ticks of that clock:
$$-(\nu_{\mathsf 2} - \nu^{\text{rec}}_{\mathsf 1 \, \leftarrow \, \mathsf 2}) \approx (\nu_{\mathsf 1} - \nu^{\text{rec}}_{\mathsf 2 \, \leftarrow \, \mathsf 1}) \approx 21.18~{\rm Hz}$$
?)
