  Locally AdS$_{d-1}\times\mathbb{R}$ uniform black strings (UBS) in the
presence of a massless scalar field are believed to avoid the onset of the
Gregory-Laflamme (GL) instability in $d\ge4$ as no tachyonic modes exist in the
spectrum of the Laplace-Beltrami operator. We present analytic and numerical
evidence of GL modes in the Lichnerowicz spectrum indicating that AdS$_{d-1}$
UBSs are classically and thermodynamically unstable at the linear level in
$d>4$. In $d=4$, we confirm that uniform BTZ$_3$ strings are indeed stable as
previously suggested. We propose that linear instabilities of black strings are
triggered if and only if a tachyonic mode exists in the Lichnerowicz spectrum.
At the end state of the instability, AdS$_{d-1}$ UBSs of finite length may
tunnel to a SAdS$_d$ black hole or converge onto a novel non-uniform AdS$_d$
black string. We conjecture that weak cosmic censorship is violated if the
non-uniform solution is an exact AdS$_d$ black funnel and compute entropy
estimates in $d>4$ as evidence.
