special relativity - Is there a time operator in quantum mechanics?

The question in the title has been asked many times on this site before, of course. Here's what I found:

1. Time as a Hermitian operator in QM? in 2011. Answer states time is a parameter.


2. Is there an observable of time? in 2012. Answer by Arnold Neumaier cites surveys in the literature, and very briefly summarizes Pauli's 1958 proof: if a time operator satisfied a CCR with the Hamiltonian, it would imply spectrum of Hamiltonian is unbounded below.


3. Why there is no operator for time in QM? in March 2013. Closed as duplicate without answer.



4. What are the Time Operators in Quantum Mechanics? in Nov 2013. Answer confusingly states there is no time operator, then describes a time operator. Then closed as duplicate.


5. What is the correct way to treat operators that has "time" in QM? in Feb 2016. Answer describes an explicit time operator, but points out that multiplication by $t$ does not yield an $L^2(\mathbb{R})$ function since norm is preserved in time.


6. Let me also link Why isn't the time-derivative considered an operator in quantum mechanics?, Why $\displaystyle i\hbar\frac{\partial}{\partial t}$ can not be considered as the Hamiltonian operator?, Time in special relativity and quantum mechanics, and Why isn't the Heisenberg uncertainty principle stated in terms of spacetime? which address slightly different questions, but contain related discussions.


7. We also have Position operator in QFT, where Valter Moretti points out that a sensible construction of coordinate operators in relativistic QFT yields the Newton-Wigner operators, but only for the spatial coordinates, since none exists for the timelike coordinate by Pauli's theorem.


8. And off of physics.se, a great resource is John Baez's website, where he has a note The Time-Energy Uncertainty Relation which mentions that by the Stone-von Neumann a time operator satisfying a CCR with the Hamiltonian would necessitate the Hamiltonian have unbounded spectrum.

Should Pauli's theorem (which is probably a corollary of Stone-von Neumann theorem), as it's referred to in some of the answers, be taken as the definitive answer? Most of the sources seem to agree.

But some of these answers seem to be ignoring an elephant: in a relativistic theory, time and space are treated on an equal footing. If there are spatial operators, surely there must be a time operator. Indeed, some of the comments in the above questions refer to chapter 1 of Srednicki's book. Let me quote the relevant passage:

One [option to combine quantum mechanics and relativity] is to demote position from its status as an operator, and render it as an extra label, like time. The other is to promote time to an operator.

Let us discuss the second option first. If time becomes an operator, what do we use as the time parameter in the Schrödinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time $T$ (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators $X^\mu(τ)$, where $X^0 = T$. Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so.

In a subsequent paragraph, he points out that promoting all the coordinates to operators, including time, is exactly what string theory is, though with the additional complication that there is an additional parameter.

So how do we reconcile these two camps? Is Pauli's theorem incompatible with relativistic considerations? Is it really a problem if your Hamiltonian's spectrum is not bounded below? Why is there no timelike Newton-Wigner operator? Does there in fact exist a relativistic quantum theory which includes an operator whose eigenvalues are time coordinates?

I feel like the existing answers on the site did not address this issue, so I'm hoping for some new answers which look at it from that perspective.