  We show that given a real number $p<1$, a positive integer $n$ and a proper
subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere
$\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the
class of Borel subsets of $\mathbb{S}^{n-1}\cap H$ equals the spherical
Lebesgue measure on $\mathbb{S}^{n-1}\cap H$, is not the $L_p$-surface area
measure of any convex body. This, in particular, disproves a conjecture from
[Bianchi, B\"or\"oczky, Colesanti, Yang, The $L_p$-Minkowski problem for
$-n<p<1$, Adv. Math. (2019)].
