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It should work when you open the PDF file in a PDF reader program like Adobe Reader.
Here's a screenshot of the web form:
Now look at the screenshot when I open the fillable PDF in Adobe Reader.
As you can see, there is a red border box on the text boxes indicating that it is a required field. The fillable PDF form cannot be submitted if these fields are blank.
In Google Chrome, there is * or red border. It just a regular text box but if the required text box is empty, you can't submit the form.
Hope that helps. Thank you.