Are Einstein's Boxes an argument for nonlocality?

(an experimental proposal)

Today I want to discuss a topic from an excellent book by Jean Bricmont: Making Sense of Quantum Mechanics which presents the best arguments for the Bohmian interpretation. Although I do not agree with this approach I appreciate the clarity of the arguments and I want to present my counter argument.

On page 112 there is the following statement: "... the conclusion of his [Bell] argument, combined with the EPR argument is rather that there are nonlocal physical effects (and not just correlations between distant events) in Nature". 

To simplify the argument to its bare essentials, a thought experiment is presented in section 4.2: Einstein's boxes. Here is how the argument goes: start with a box B and a particle in the box, then cut the box into two half-boxes B1 and B2. If the original state is \(|B\rangle\), after cutting the state it becomes:

\(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) 

Then the two halves are spatially separated and one box is opened. Of course the expected thing happens: the particle is always found in one of the half-boxes. Now suppose we find the particle in B2. Here is the dilemma: either there is action at a distance in nature (opening B1 changes the situation at B2), or the particle was in B2 all along and quantum mechanics is incomplete because \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\) does not describe what is going on. My take on this is that the dilemma is incorrect. Splitting the box amounts to a measurement regardless if you look inside the boxes or not and the particle will be in either B1 or B2.  

Here is an experimental proposal to prove that after cutting the box the state is not \(\frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\):

split the box and connect the two halves to two arms of a Mach-Zehnder interferometer (bypassing the first beam splitter). Do you get interference or not? I say you will not get any interference because by weighing the boxes before releasing the particle inside the interferometer gives you the which way information.

If we do not physically split the box, then indeed \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\), but if we do physically split it \(|B\rangle \neq \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a hidden assumption in Einstein's boxes argument: realism which demands non-contextuality. Nature and quantum mechanics is contextual: when we do introduce the divider the experimental context changes. 

Bohmian's supporters will argue that always \(|B\rangle = \frac{1}{\sqrt{2}}(|B_1\rangle+|B_2\rangle)\). There is a simple way to convince me I am wrong: do the experiment above and show you can tune the M-Z interferometer in such a way that there is destructive interference preventing the particle to exit at one detector.