Consider the following physical situation. Two position detectors are located next to each other, each one carries only YES/NO information on whether the particle hit that detector. A particle, let's say in a Gaussian state, is headed toward the two detectors. The width of the wavefunction of the incident particle is such that it would overlap with both detectors upon reaching the detection surface.

My question is, what would be (heuristically) the post-measurement state of this system, disregarding wavefunction collapse? For spin measurements it is common to write something like

$$(\alpha |\uparrow\rangle + \beta |\downarrow\rangle)\otimes |ready\rangle \to \alpha |\uparrow\rangle |\uparrow \text{detected} \rangle + \beta |\downarrow \rangle |\downarrow \text{detected} \rangle $$

where the first state in each tensor product represents the measured particle, and the second represents the state of the detector/environment. But for a continuous variable like position the answer is not so clear to me.