Partially true. If we could manufacture 100km single-crystal graphene (e.g. carbon fiber but sheets) then that would be sufficiently strong, and to prevent unravelling you could literally use something as weak and heavy as scotch tape to hold different ribbons of the graphene together. More ribbons, more tape, bigger and stronger cable.

The problem is the single-crystal manufacturing, which we can't do. Without it, we need a high-tensile-strength glue to hold the strands we can manufacture together. No such glue exists, especially not one that could connect via polar and/or van-der-walls bonds to the graphene.

Ergo, no space elevator, yet.