Thank you for the nice formatting the mauscript AG11709!

To address the specific questions:

1. Headings. We find that with this relativley short article, and the way it
is structured at the moment, a clear use of section headings would not be too
helpful for the reader. Therefore, we would like to keep the article structure
as it is, without the headings (apart from the Acknowledgement section).

2. Verification of figure reference in line 93 of the proof:
Please replace "Figs. 1(b) and (d)" with "Figs. 1(b) and 1(d)".

3. Update of e-print for reference [35]:
The work is currently under review, and we don't have a reference yet.

FQ: The funding body for the LKY postdoctoral fellowship has no fundref entry,
but see next item.

4. Funding provider line 224/225:
Please revert the reference to the full name (required by funding agency):
"...acknowledges support by the Lee Kuan Yew Postdoctoral Fellowship."