Legally, yes — but the demand becomes hollow. Both halves of that sentence matter, so take them one at a time.
The legal position, as the chapter states it. Rights are protected by law: if someone’s rights are violated, even by the government, they can go to court for help. Duties, on the other hand, are moral and social responsibilities — “we are expected to follow them, but not doing so may not always lead to legal action.” The chapter does not say your Fundamental Rights are switched off if you fail your Fundamental Duties. A person who litters does not thereby lose the right to a fair trial.
The practical position. Now look at what a right actually consists of:
My right to speak is worth something only if others carry the duty to listen instead of shouting me down.
My right to a clean environment is worth something only if others carry the duty to not throw wrappers on the ground.
My right to education is worth something only if adults carry the duty to send children to school instead of to work.
My right to be respected is worth something only if others carry the duty to respect people they find different.
Why it happens: every right is a claim on somebody else’s behaviour. If I demand my rights and refuse my duties, I am asking others to do for me what I will not do for them. That cannot be a rule for everybody at once — and a rule that cannot apply to everybody is not a principle, it is a privilege. This is why the chapter says rights and duties are not opposites but go together, and why it opens with Mahatma Gandhi: “The true source of rights is duty.”
The short answer to write: we can demand them, because the Constitution does not make our rights conditional on our duties — but we cannot enjoy them, because the duties of others are what our rights are made of.