  We study a Bishop-Phelps-Bollob\'as type property for Banach space operators
introduced by Dantas (2017). In that paper there is a local and a global
version of a natural property which is somewhat similar but simpler compared to
the Bishop-Phelps-Bollob\'as type property for operators studied in Acosta et
al. (2008). Here we characterize the mentioned local property in the setting
with strictly convex domain spaces and compact operators. We show that the
local property implies that the domain space has strong convexity properties.
