M-diagram basis of the Specht module for $(n,n,n)$

Motivated by the M-diagrams defined by Tymoczko, we show that these locally non-crossing $\mathfrak{sl}_3$-webs form a basis of the Specht module for the partition $(n,n,n)$. They further admit a unitriangular base change to both the polytabloid basis and the non-elliptic web basis.