  In Property Testing, proximity-oblivious testers (POTs) form a class of
particularly simple testing algorithms, where a basic test is performed a
number of times that may depend on the proximity parameter, but the basic test
itself is independent of the proximity parameter. In their seminal work,
Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that
allow constant-query proximity-oblivious testing in the bounded-degree model
are precisely the properties that can be expressed as a generalised subgraph
freeness (GSF) property that satisfies the non-propagation condition. It is
left open whether the non-propagation condition is necessary. Indeed, calling
properties expressible as a generalised subgraph freeness property GSF-local
properties, they ask whether all GSF-local properties are non-propagating. We
give a negative answer by exhibiting a property of graphs that is GSF-local and
propagating. Hence in particular, our property does not admit a POT, despite
being GSF-local. We prove our result by exploiting a recent work of the authors
which constructed a first-order (FO) property that is not testable [SODA 2021],
and a new connection between FO properties and GSF-local properties via
neighbourhood profiles.
