Does the stay in country of entry have to be longest in a multiple entry Schengen visa?

Where and how to apply

You must lodge the application for a Schengen visa at the Consulate of the country that you intend to visit, or – if you intend to visit more than one Schengen State, the Consulate of the country where you will spend the longest period.

If you intend to visit several Schengen States and the stays will be of equal length, you must apply at the Consulate of the country whose external borders you will cross first when entering the Schengen area.

Different FAQ documents from different but still official places put this differently:

Frequently Asked Questions

You must lodge the application for a Schengen visa at the Consulate of the country that you intend to visit, or – if you intend to visit more than one Schengen State, the Consulate of the country of your primary destination (i.e. main purpose of stay or longest stay).

Note the subtle difference: "main purpose of stay" is added.

Here's another, stil from the EEAS site General Schengen Visa requirements:

You are not free to choose at which mission of a Schengen state to apply.

You have to apply at the mission of the country where your main destination/ longest stay will be or if it is difficult to determine (e.g. cruises, bus tours) you must apply from the mission of the first point of entry.

Again main destination / longest stay appears but now not only the equivalent length allows for applying at first point of entry but also "difficult to determine".

All in all, if you just go to visit, organized by yourself and plan to stay somewhere longer than other places then file there. If it's the same everywhere, say, one week, apply at the initial country -- but if your route is [Non schengen airport] - [Schengen airport in country A] - [Schengen airport in country B] then the first country is A and if you then spend a week on your own volition in B, C, D, E expect them to at least give extra scrunity of whether you are trying to game the system by filing in A. Just spend eight days in, say, C instead of one week and file for a visa there.

In my personal opinion this is a really moronic system. Obviously they didn't want to put the burden on visas on the first cross countries because then those with large airports would get an unduly load. Instead there should be an electronic system where you start your application and it simply sends your application to the Schengen country with the fewest applications at the moment (that have a visa office at your residency)...

You should apply for schengen visa at italian consulate since you intend to stay there longest.

Please follow these rules according your trip. You must apply for schengen visa according to these official rules.

https://eeas.europa.eu/sites/eeas/files/frequently_asked_questions_en.pdf

You must lodge the application for a Schengen visa at the Consulate of the country that you intend to visit, or – if you intend to visit more than one Schengen State, the Consulate of the country of your primary destination (i.e. main purpose of stay or longest stay).

If you intend to visit several Schengen States and the stays will be of equal length, you must apply at the Consulate of the country whose external borders you will cross first when entering the Schengen area. As a general rule, you must apply for a visa at the Consulate with territorial competence for the country in which you legally reside.

https://ec.europa.eu/home-affairs/what-we-do/policies/borders-and-visas/visa-policy/apply_for_a_visa_en

The application must, in principle, be submitted to the Consulate at least 15 days before the intended journey and cannot be lodged earlier than three months before the start of the intended journey. You may have to book an appointment before lodging the application.