Travel outside Common Travel Area while application for naturalisation or citizenship is in progress

Hi, hoping to get an answer for this query.
My wife is an EU citizen and already has a permanent residence card/status. She is in the middle of applying for naturalisation/UK citizenship (Already completed her application, made the payment (and I am assuming the application has been submitted?) but has not booked the appointment with UKVCAS yet). We already had plans to travel abroad for 03 weeks for a holiday in September and then again in November where these trips have already been booked and paid for.

Before booking the appointment with UKVCAS & I was browsing through some articles online to check whether she would get her passport back after the appointment that we cam across this article third party weblink removed by moderator
where writes states that
A key benefit of the new system is that applicants won’t have to send off their original documents (including their passport) and so won’t run the risk of their documents being misplaced by the Home Office or lost in the postal system. However, applicants should not use their passport to travel outside the Common Travel Area (the UK, Republic of Ireland, Channel Islands and Isle of Man) while their application is being processed, as the Home Office has stated that their application will be treated as withdrawn. if they do leave the UK while their application is being processed, as there is a risk this will lead to the Home Office treating the application as withdrawn.

This confused us a little bit initially but I have found the below on Home Office's official guide at the below link:
https://www.gov.uk/guidance/immigration ... 1withdrawn


34J. The proof of identity provided under paragraph 34(5) will be returned to the applicant whilst their application is being considered, unless the Secretary of State considers it necessary to retain it. Where the Secretary of State has retained an applicant’s proof of identity and the applicant requests the return of their passport for the purpose of travel outside the common travel area, the application for leave shall, provided it has not already been determined, be treated as withdrawn on the date that request is received by the Home Office.
34K. Where proof of identity provided under paragraph 34(5) has been returned to the applicant pending a decision on their application for leave to remain and the applicant travels outside the common travel area their application for leave to remain shall, provided that it has not been determined, be treated as withdrawn on the date that the applicant left the common travel area.

My Question is that above 02 paragraphs only refer to an applicant applying for "leave to remain" so would this rule apply to her? if yes, is there an alternative/solution for this?

Thanks a lot in advance

You will retain your original documents including Passport/ID card and PR card, and you are free to travel. Be aware that UKVI may still ask further questions or request additional documents so it would be useful to have someone check your mail during your absence. Naturalisation is not an immigration application some of which require you to stay in the country. So doesn't apply to your wife.

@alterhase58
Thank you for answering the question and thanks for the advice too. I'll get someone to check our mail while we are away.