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@ L Spice: I think it holds for $p=2$. you are right, the reference I mentioned does assume $p=2$, however in Labesse and Langlands, L-Indistinguishability for SL(2), $p$ is arbitrary. Also if you believe that the size of the $L$-packet is in bijection with the component group of the centralizer of the associated projective Weil-Deligne representation it is an exercise to compute that the only possible sizes are of 1,2 or 4.
