What's the status of Arthur's announced classification for GSp(4)?

This question is answered pretty definitively by the following recent paper:

Gee, Toby; Taïbi, Olivier, Arthur’s multiplicity formula for $\mathrm{GSp}_4$ and restriction to $\mathrm{Sp}_4$, Journal de l'École polytechnique — Mathématiques, Volume 6 (2019), p. 469-535.

Abstract: "We prove the classification of discrete automorphic representations of GSp4 explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp4 and Sp4."

His monograph has appeared, albeit under a slightly different name:

James Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Colloquium Publications 61 (2013) 590 pp, AMS bookstore.

A student of Arthur, Bin Xu, has worked on the general problem of classifying automorphic representations for $GSp(2n)$. See his webpage here, or better:

Bin Xu, Endoscopic Classification of Representations of $GSp(2n)$ and $GSO(2n)$, PhD thesis, University of Toronto (2014) (http://hdl.handle.net/1807/68169).

and

Bin Xu, L-packets of quasisplit $GSp(2n)$ and $GO(2n)$, Math. Ann. 370 (2018) pp 71–189, doi:10.1007/s00208-016-1515-x, arXiv:1503.04897.