Modern discussion has narrowed the field without settling it, and the four names that recur are jacinth, understood as the hyacinth variety of zircon, amber, tourmaline and opal. Each is argued for on a different kind of ground, and the difference between those grounds is part of why the argument does not resolve.
The case for jacinth runs largely through the translation tradition itself, since several of the ancient renderings point that way. That makes it strong on continuity and weak on independence: it is the answer the chain of translators arrived at, and the chain is the thing in question. Amber is argued from the material's demonstrable presence in the ancient Levant through the Baltic trade and from etymological readings of the Hebrew, though amber is a fossil resin rather than a mineral and sits oddly in a list otherwise composed of stones. Tourmaline and opal are proposed on comparison with other ancient Near Eastern stone lists and on availability, and both meet the same objection: availability establishes that a stone could have been used, not that this particular word named it.
None of the four has produced the kind of evidence that would end the discussion. The atlas therefore records the question as open rather than presenting a preferred candidate as though the matter were closer to settled than it is, and it carries the open question against this study rather than a conclusion. What would close it is not exotic: one securely dated use of leshem outside the breastplate lists, in a context that identifies a physical object, would very likely be enough.