"Engaged" is one of those terms that come from MBA school, and has no bearing whatsoever on how much an employee contributes to your company. The most "engaged" employees are the ones who play the politics game the most, and say everything the higher ups want to hear, regardless of if it's right or not. They're the "yes men/women". And of course they want to be there in person, because their whole existence is based around being seen by the right people, talking to as many people as possible, etc.
Productive is what you actually should want in an organization, not necessarily "engaged". And productive employees are the ones who spend the least time socializing in the office, and who want the least distractions. They are the ones who want to just get the work done. These people are better off at home, assuming they still have access to all the tools they need for their job.
Note that I'm using engaged in quotes, because real engagement is often the opposite of what is measured when people with management degrees talk about "engagement". The employees with real engagement are the ones who tell the boss no when it's appropriate, are the ones who get frustrated with broken processes, and won't tolerate idiots on their team. They're the ones who truly want what's best for the company, but because they point out the flaws in the company, don't want to waste time gossiping in the break room, and are rarely seen because they're too busy doing the actual work, they are labelled by the MBA types as not being "engaged".