Inequalities for Extensions of Unbounded Operators

Question

Background

I am trying to understand the (non-)tension between the following definitions/propositions. I will always assume I am working with states in a Hilbert space H and operators on that Hilbert space; and I use physics braket notation. I am following the discussion in Section 3 of https://arxiv.org/pdf/1803.04993.pdf. Let me try to summarize the argument and notation:

1. A self-adjoint operator $P$ is called positive if $\langle \chi |P| \chi \rangle \geq 0$, denoted $P \geq 0$.
2. If $P$ and $Q$ are bounded, so that they're defined for the whole Hilbert space $H$, we write $P\geq Q$ if $P-Q \geq 0$. This is apparently equivalent to $$ \frac{1}{s+P} \leq \frac{1}{s+Q} \qquad (*) $$ for all $s > 0$.
3. If $P$ and $Q$ are densely defined unbounded non-negative operators, then we take the formula $(*)$ as the definition of $P \geq Q$. This is nice because the unbounded operators do not have to be defined on the same dense subspaces of $H$, but the bounded operators above are defined for all $\chi \in H$. This last point will be key to the confusion.

Later in the paper, we concern ourselves with extensions of unbounded operators:

4. Let $X$ and $Y$ unbounded operators on $H$. $X$ is an extension of $Y$ if $X$ is defined whenever $Y$ is defined, and they agree on the common subset on which they are defined. In our case of concern, both $X$ and $Y$ are defined on dense subsets of $H$ and $X$ is an extension of $Y$ as above.
5. Finally, it is shown that if $X$ is an extension of $Y$, that $X^\dagger X \leq Y^\dagger Y$.

Question

The tension is as follows: Naively, one would expect that if the unbounded operators $X$ and $Y$ agree on a dense set of $H$, then the operators $P = Y^\dagger Y$ and $Q = X^\dagger X$ agree on a dense set of $H$, and then the bounded operators in $(*)$ agree on a dense set, and so agree everywhere. This is presumably false if the inequality is going to make sense.

Is there any way to see where the logic has failed? Or maybe I am just confused by notation and I need to define things more carefully through closure of graphs? In the case of interest in the paper, the operators $X$ and $Y$ are Tomita operators, and are essentially defined to be closed following the definitions in the paper, so the adjoints should be densely defined?