When you have doubts about what your employee may be doing online, there is no doubt that you need to act immediately, whether your employee requests a cover or not. There are several ways to find out what employees are doing online, ranging from simply spending more time with them to obtaining equipment that actually displays what your employees have on their monitors.
As for the screen cover, it does sound somewhat suspicious that your employee would request this item out of the blue. However, if his job includes handling sensitive information, there may be a legitimate business reason for the request. In addition, some of these covers reduce glare, and his request may be as simple as that.
If you want to shed more light on this problem, you should meet with this employee to discuss his reasons for this request. Before agreeing to a filter for his monitor, it sounds like you should monitor his work more carefully.
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