Heesch's problem in the Euclidean plane: a tile with Heesch number $\ge7$, or a bound on finite Heesch numbers

The Heesch number of a plane tile $T$ is the maximum number of times $T$ can be completely surrounded by successive coronas of congruent copies of itself without the copies tiling the whole plane: it is the largest $k$ for which there exist $k$ concentric coronas of congruent copies of $T$ around a central copy (each corona a ring of tiles covering the boundary of the previous configuration, with no overlaps and no gaps adjacent to the covered region), where $T$ nonetheless does not admit a tiling of the plane. Heesch's problem, in the EUCLIDEAN plane, asks: which positive integers occur as Heesch numbers of plane tiles, and is the supremum of the finite Heesch numbers bounded?

Acceptance. FULLY RESOLVES / headline ADVANCES: an explicit plane tile (polyform or marked / edge-decorated shape, with exact geometry) of Heesch number $\ge7$, certified by a valid $7$-corona surround computation together with a proof that the tile does not tile the plane - the first improvement on the record $6$. ADVANCES: a rigorous structural bound on the finite Heesch numbers of Euclidean plane tiles (any finite upper bound would be a breakthrough), or a proof that some specific integer $>6$ is (or is not) attainable. ADVANCES: verified new Heesch numbers for natural tile families. Heuristic surrounds without a certified non-tiling proof do NOT qualify.