Applying for Schengen (France) visa in US (Proof of socio-economic status)

I am an Indian citizen with a US Green Card and I reside and work in the USA as a university professor. I am planning on applying for a short stay Schengen visa (France) for tourism. The French Consulate with jurisdiction is Chicago.

To apply for a short stay Schengen visa the following documents are mentioned under "proof of socio-economic status":

• letter of employment in the U.S. confirming your salary, hire date, position and dates of vacation (business registration if self employed) and your last 3 pay stubs (unless downloaded, please bring a photocopy)

• or letter from your school or university confirming your enrollment status, attendance and expected graduation date

• For a business trip: please present a letter from your employer or University regarding the purpose of your trip and an official invitation from your hosting company in France

Now, I wish to not ask my university for a letter since I do not need them to know where I am vacationing during "off school" season. I also have a prior long stay French visa (chercheur scientifique) in my Indian passport (2014-2015). I also have sufficient funds and of course plan on having travel arrangements etc. made before I apply for a visa.

This lack of "letter from employer": would it play negatively in the visa process?