  Many structural glasses feature static and dynamic mechanical properties that
can depend strongly on glass formation history. The degree of universality of
this history-dependence, and what it is possibly affected by, are largely
unexplored. Here we show that the variability of elastic properties of simple
computer glasses under thermal annealing depends strongly on the strength of
attractive interactions between the glasses' constituent particles -- referred
to here as glass `stickiness'. We find that in stickier glasses the stiffening
of the shear modulus with thermal annealing is strongly suppressed, while the
thermal-annealing-induced softening of the bulk modulus is enhanced. Our key
finding is that the characteristic frequency and density per frequency of soft
quasilocalized modes becomes effectively invariant to annealing in very sticky
glasses, the latter are therefore deemed `thermo-mechanically inannealable'.
The implications of our findings and future research directions are discussed.
