Google sheets, hide/remove data validation dropdown..unless another cell is not empty?

I have a column that has dropdown list with data validation, and its applied infinitely down that column. So I have lots of cells with the little dropdown arrow which is annoying.

Is there a way to remove that dropdown arrow....IF say "Date" or "Name" column is empty? I want the drop to to show only when I have started inputting data to that row, I will never start with the drop down, I start with date. So ideally, as soon after entering the date, the dropdown arrow appears.

Can that be done? Thanks!

You can remove the dropdown arrows by unchecking Show Dropdown List in Cell on Data Validation dialog:

This will show the dropdown list only when you start entering data to the cell.