No, based on the information you provided, it does not sound like you have a NY filing obligation.
New York has a Convenience of the Employer Rule. That means, if your employer requires you to work from home, your wages are not taxable in NY. If you are allowed to work from home, as a convenience to you as the employee, then the wages are taxable in NY, whether or not you work there physically.
It sounds like, since there is no space available for you, you are required to work from home. Thus, the wages are not taxable to NY, and instead they are taxable to UT.