I was going through example 2.10 in Griffiths(Introduction to Electrodynamics 4th edition), everything seemed okay, but at last he makes an assertion, the proof for which I am not able to understand.

Nope: As we'll see in the uniqueness theorems of Chapter 3, electrostatics is very stingy with its options; there is always precisely one way -- no more -- of distributing the charge on a conductor so as to make the field inside zero. Having found a possible way, we are guaranteed that no alternative exists, even in principle.

I searched for uniqueness theorems in Chapter 3, and I got two of them, but none of them seem to directly imply that there is always precisely one way- no more--of distributing the charge on a conductor so as to make the field inside zero.

First uniqueness theorem states that:

The solution to Laplace's equation in some volume $V$ is uniquely determined if $V$ is specified on the boundary surface S.

and the second one states that:

In a volume V surrounded by conductors and containing a specified charge density p, the electric field is uniquely determined if the total charge on each conductor is given  (Fig. 3.6). (The region as a whole can be bounded by another conductor, or else unbounded.)

My question: How is the assertion made in example 2.10 justified by the uniqueness theorems?

I was going through example 2.10 in Griffiths  (Introduction to Electrodynamics 4th edition), everything seemed okay, but at last he makes an assertion, the proof for which I am not able to understand.

Nope: As we'll see in the uniqueness theorems of Chapter 3, electrostatics is very stingy with its options; there is always precisely one way -- no more -- of distributing the charge on a conductor so as to make the field inside zero. Having found a possible way, we are guaranteed that no alternative exists, even in principle.

I searched for uniqueness theorems in Chapter 3, and I got two of them, but none of them seem to directly imply that there is always precisely one way- no more--of distributing the charge on a conductor so as to make the field inside zero.

First uniqueness theorem states that:

The solution to Laplace's equation in some volume $V$ is uniquely determined if $V$ is specified on the boundary surface $S$.

and the second one states that:

In a volume $V$ surrounded by conductors and containing a specified charge density $p$, the electric field is uniquely determined if the total charge on each conductor is given  (Fig. 3.6). (The region as a whole can be bounded by another conductor, or else unbounded.)

My question: How is the assertion made in example 2.10 justified by the uniqueness theorems?