Corneal transplantation succeeds far more often than other transplants, and the reasons trace back to two facts: the cornea has no blood vessels and the graft survives only if its endothelial pump is intact. These two facts let us sort the four options quickly.
The endothelium is a single layer that keeps the cornea dehydrated and clear. Before a graft is used, eye banks look at this layer with a specular microscope, which photographs the endothelial cells and gives a count and a measure of cell shape. A low count predicts graft failure, so this is the central quality check. That makes the specular microscopy statement the true one.
The idea that the entire globe is stored in tissue culture is incorrect. Eye banks remove a corneoscleral button and keep it in storage media; the intact eyeball is only used for short-term moist chamber preservation. The age limit statement is also false, since corneas from donors over 60 are routinely transplanted as long as the endothelial count is adequate. Finally, HLA matching is not mandatory for a standard graft because the avascular, immune-privileged cornea tolerates mismatched tissue; matching is reserved for vascularised high-risk beds.
So the only accurate statement concerns endothelial cell counting by specular microscopy.
\[\boxed{\text{Specular microscopy assesses endothelial cell count}}\]