Question Regarding DPDA acceptance inspired from GATE 2005 Question

Question

The following question is a modified version of this question https://gateoverflow.in/3785/gate2005-it-38, in this GATE question they have NPDA, but I'm asking about DPDA

Let D be a deterministic push-down automaton (DPDA) with exactly one state, q, and exactly one symbol, Z, in its stack alphabet. State q is both the starting as well as the accepting state of the DPDA. The stack is initialized with one Z before the start of the operation of the DPDA. Let the input alphabet of the DPDA be Σ. Let L(P) be the language accepted by the DPDA by reading a string and reaching its accepting state. Let N(P) be the language accepted by the DPDA by reading a string and emptying its stack. 
Which of the following statements is TRUE?

1. L(P) is necessarily Σ* but N(P) is not necessarily Σ*.
2. N(P) is necessarily Σ* but L(P) is not necessarily Σ*.
3. Both L(P) and N(P) are necessarily Σ*.
4. Neither L(P) nor N(P) are necessarily Σ*.

Comments

For me, I think, acceptance by empty stack is not necessarily guranteed because Σ* will fail to satisy prefix property, but what about acceptance by final state.

Answer 1

Option 1 is correct.

L(P) is necessarily Σ* as initial state is final state and null move is not needed for acceptance by final state.

but N(P) is not necessarily Σ* because language Σ* has prefix property which will make symbol move and null move both necessary.