  The lineshapes of specific production experiments of the exotic state such as
$X(3872)$ with $J^{PC}=1^{++}$ quantum numbers involving triangle singularities
have been found to become highly sensitive to the binding energy of weakly
bound states, thus offering in principle the opportunity of benchmark
determinations. We critically analyze recent proposals to extract accurately
and precisely the $X(3872)$ mass, which overlook an important physical effect
by regarding their corresponding production lineshapes as a sharp mass
distribution and, thus, neglecting the influence of initial nearby continuum
states in the $1^{++}$ channel. The inclusion of these states implies an
effective cancellation mechanism which operates at the current and finite
experimental resolution of the detectors so that one cannot distinguish between
the $1^{++}$ bound-state and nearby $D \bar D^*$ continuum states with the same
quantum numbers. In particular, we show that the lineshape for resolutions
above 1 MeV becomes rather insensitive to the binding energy unless high
statistics is considered. The very existence of the observed bumps is a mere
consequence of short distance correlated $\bar D D^*$ pairs, bound or unbound.
The cancellation also provides a natural explanation for a recent study
reporting missing but unknown decay channels in an absolute branching ratio
global analysis of the $X(3872)$.
