They made it less than 2 miles. Clearly it went wrong shortly after loss of visual references. As mentioned, simple things can cause a serious UA shortly after takeoff.
Something as simple as a stuck down cyclic FTR can cause complete chaos with even a competent crew shortly after departure. An inadvertent SAS release with no references can be catastrophic at low alt.
The FD is very smart post phase 4 and should prevent an inadvertent descent except in a few rare conditions.
Whatever the cause, the MPFR will have most of the answers. RIP.